Quentin Berger
email: quentin.berger@sorbonne-universite.fr

Matthias Birkner
email: birkner@mathematik.uni-mainz.de

Linglong Yuan
email: linglong.yuan@liverpool.ac.uk

Collective vs. individual behaviour for sums of i.i.d. random variables: appearance of the one-big-jump phenomenon

Keywords: Large deviation, Local large deviation, Extended regular variation, Intermediate regular variation, Phase transition, One-big-jump phenomenon, α-stable law

This article studies large and local large deviations for sums of i.i.d. real-valued random variables in the domain of attraction of an α-stable law, α ∈ (0, 2], with emphasis on the case α = 2. There are two different scenarios: either the deviation is realised via a collective behaviour with all summands contributing to the deviation (a Gaussian scenario), or a single summand is atypically large and contributes to the deviation (a one-big-jump scenario). Such results are known when α ∈ (0, 2) (large deviations always follow a one big-jump scenario) or when the random variables admit a moment of order 2 + δ for some δ > 0. We extend these results, including in particular the case where the right tail is regularly varying with index -2 (treating cases with infinite variance in the domain of attraction of the normal law). We identify the threshold for the transition between the Gaussian and the one-big-jump regimes; it is slightly larger when considering local large deviations compared to integral large deviations. Additionally, we complement our results by describing the behaviour of the sum and of the largest summand conditionally on a (local) large deviation, for any α ∈ (0, 2], both in the Gaussian and in the one-big-jump regimes. As an application, we show how our results can be used in the study of condensation phenomenon in the zero-range process at the critical density, extending the range of parameters previously considered in the literature.

1

The general theme of the present paper is to study the interplay between S n := n i=1 ξ i and M n := max{ξ 1 , ξ 2 , . . . , ξ n }, when one of them is atypically large.

We assume that ξ is in the domain of attraction of an α-stable law, with α ∈ (0, 2], i.e. that there exist sequences (a n ) n≥1 , (b n ) n≥1 such that (S n ) n≥1 satisfies

S n -b n a n d ---→ n→∞ S α , (1.1)
where S α is an α-stable random variable and d -→ denotes convergence in distribution. From Feller [Fel08, Ch. XVII.5, Thm. 2], a necessary and sufficient condition is the following:

• If α ∈ (0, 2): there is a slowly varying function L(•) such that F (x) ∼ pL(x)x -α , F (-x) ∼ qL(x)x -α as x → ∞ ,

(1.2) with p, q ≥ 0, p + q = 1; if p = 0, we interpret (1.2) as F (x) = o(L(x)x -α ) and similarly if q = 0. Note that in this case we have that F (x) + F (-x) ∼ L(x)x -α as x → ∞.

• If α = 2: ξ has a finite expectation, that we denote by µ = E[ξ], and the truncated variance

σ 2 (x) := E (ξ -µ) 2 1 {|ξ-µ|≤x} , x ≥ 0 (1.3)
is slowly varying as x → ∞. Note that this contains the case of a finite second moment, i.e. lim x→∞ σ 2 (x) = E[(ξ -µ) 2 ] =: σ 2 .

In the case α = 2, let us recall the fact that (see [START_REF] Feller | An introduction to probability theory and its applications[END_REF]Ch. IX.8, Eq. (8.5)])

F (x) + F (-x) = o(x -2 σ 2 (x)), as x → ∞.

(1.4)

The normalising sequence (a n ) n≥1 verifies, as n → ∞

L(a n )a -α n ∼ n -1 if α ∈ (0, 2); σ 2 (a n )a -2 n ∼ n -1 if α = 2 , (1.5)
and the centering sequence (b n ) n≥1 is given by

b n = 0 if α ∈ (0, 1); b n = nE ξ1 {|ξ|≤an} if α = 1; b n = nE[ξ] if α ∈ (1, 2] . (1.6)
Note that we can also replace (1.5) by L(a n )a -α n ∼ an -1 , σ 2 (a n )a -2 n ∼ an -1 in the different cases for any a > 0. The value of a only changes the law of S α by a dilation factor.

The one-big-jump phenomenon

The "one-big-jump phenomenon" asserts that the large deviation event of having S n unusually large may be realised essentially thanks to exactly one of the variables ξ 1 , . . . , ξ n being very large. This is captured in the large deviation behaviour of the probability

P(S n -b n ≥ x n ) ∼ P(S n -b n ≥ x n , M n ≥ x n ) ∼ P(M n ≥ x n ) as n → ∞ , (1.7)
where x n is a sequence going to infinity. We also refer to Theorem 2.10 and Corollary 3.2 below for a more detailed interpretation of (1.7). Finding conditions on the distribution of ξ and on the sequence (x n ) n≥1 for (1.7) to hold has a long story, in particular within the class of subexponential distributions1 , see e.g. [CH89; Nag79; MN98] or [START_REF] Denisov | Large deviations for random walks under subexponentiality: the big-jump domain[END_REF] for a more recent reference. The one-big-jump phenomenon occurs in a large deviation regime, i.e. when P(S n -b n ≥ x n ) → 0, which is equivalent to lim n→∞ xn an = ∞, so we will focus on this regime henceforth.

Example 1.1. As a first example, consider a centered random variable ξ, i.e. µ = E[ξ] = 0, with finite variance σ 2 = Var(ξ) that verifies F (x) := P(ξ > x) ∼ L(x)x -β as x → ∞ ,

(1.8)

for some β ∈ (2, ∞) and some slowly varying function L(•). In this case, Nagaev [Nag79, Thm. 1.9] shows that if in addition E[|ξ| 2+δ ] < ∞ for some δ > 0, then for any sequence (x n ) n≥1 such that x n ≥ √ n, we have

P(S n ≥ x n ) = (1 + o(1))Φ( xn σ √ n ) + (1 + o(1))nF (x n ), n → ∞ (1.9)
where Φ is the tail probability function of the standard normal distribution. Let us set

γ n := x 2 n 2σ 2 n - β 2 -1 log n - 1 2 (β -1) log log n + log L( n log n) ,
so that in particular x 2 n ∼ σ 2 (β -2)n log n if γ n = o(log n) (which is the most interesting case). Then, using the asymptotics Φ(t) ∼ 1 t √ 2π e -t 2 /2 as t → ∞, a straightforward (but tedious) calculation gives that if lim n→∞ γ n = γ ∞ ∈ [-∞, ∞], we obtain

lim n→∞ P(S n ≥ x n ) nF (x n ) = 1 + c β,σ e -γ∞ , with c β,σ = σ β √ 2π (β -2) (β-1)/2 .
In particular, the one-big-jump phenomenon (1.7) holds if and only if lim n→∞ γ n = ∞. We will discuss how the condition that E[|ξ| 2+δ ] < ∞ for some δ > 0 (which is a condition on the left tail) and also the regular variation condition (1.8) for the right tail can be weakened, see [START_REF] Rozovskii | Probabilities of large deviations of sums of independent random variables with common distribution function in the domain of attraction of the normal law[END_REF]Thm. 6] or Theorem 2.1 below.

Local versions of (1.7), known as local large deviations in the one-big-jump regime, have also been studied, see [Ber19a; CD19; DDS08; Don89; Don97]. One usually needs to make stronger assumptions on the distribution: with reference to Example 1.1, a local version of (1.8) has been considered: P(ξ = x) ∼ βL(x)x -(1+β) with β > 2 as x → ∞, see e.g. [START_REF] Berger | Notes on random walks in the Cauchy domain of attraction[END_REF][START_REF] Doney | A large deviation local limit theorem[END_REF][START_REF] Doney | One-sided local large deviation and renewal theorems in the case of infinite mean[END_REF] or Example 2.14 below.

Observe in Example 1.1 that for distributions in the domain of attraction of the normal law which have a suitably heavy right tail, depending on the magnitude x n ≫ √ n of the deviation one studies, collective or one-big-jump behaviour can occur; in fact, it is possible that both effects contribute on an equal footing. Moreover, the changeover regime can be characterised precisely. The main new results of this paper, presented in the following Section 2, show that this situation (both integral and local) occurs for a wide class of (necessarily subexponential) distributions in the domain of attraction of the normal law. Specifically, we sharpen the profile of a result by Rozovskii [START_REF] Rozovskii | Probabilities of large deviations of sums of independent random variables with common distribution function in the domain of attraction of the normal law[END_REF], which shows how the Gaussian term and the one-big-jump term emerge in the large deviation probability P(S nb n ≥ x n ) in the case α = 2: We (partly) re-prove this under weaker assumptions in a concise way and extract sharp conditions for the changeover between collective and one-big-jump behaviour from it; we also establish the same result in the local setting, generalising Doney [START_REF] Doney | A large deviation local limit theorem[END_REF][START_REF] Doney | A local limit theorem for moderate deviations[END_REF]. Furthermore, we make the behaviour of the individual summands in the different cases explicit, corroborating the intuition behind the computations.

For completeness, we discuss in Section 3 the case of attraction to a stable law with index α < 2. This is well understood and in some sense "cleaner": Large deviations are always realised by the one-big-jump behaviour, i.e. the latter occurs if and only if x n /a n → ∞ (for the sufficiency of the criterion see [START_REF] Daren | Large deviation probabilities for sums of random variables with heavy or subexponential tails[END_REF][START_REF] Sergey | Large deviations of sums of independent random variables[END_REF][START_REF] Berger | Notes on random walks in the Cauchy domain of attraction[END_REF] and references therein; we complement this by proving necessity). We will comment on applications of our results, especially to the zero-range process, in Section 4.

2 Main results: One-big-jump vs collective behaviour in the domain of attraction of the normal law

Consider the case where ξ is in the domain of attraction of the normal law, that is α = 2.

Recall from (1.3) that it corresponds to the truncated variance σ 2 (x) = E[(ξ -µ) 2 1 {|ξ-µ|≤x} ] defined in (1.3) being a slowly varying function. This case is interesting since there is an interplay between the one-big-jump scenario and the Gaussian scenario, as seen in Example 1.1. We will need to make some assumption on the right tail of the distribution, which in particular implies subexponentiality (but is weaker than having a regularly varying tail), see (2.11) and the comments and examples below it.

Before we turn to our new results, we recall and discuss a result by Rozovskii [START_REF] Rozovskii | Probabilities of large deviations of sums of independent random variables with common distribution function in the domain of attraction of the normal law[END_REF], which provides some sharp large deviation estimates in that case, and the assumption required is weaker than E[|ξ| 2+δ ] < ∞ assumed in [Nag79, Thm. 1.9]. Afterwards, in Section 2.2 (for integral large deviations) and Section 2.3 (for local large deviations) we provide additional (and extended) results that help us understand the transition from the Gaussian to the one-big-jump regime, especially in the case that E[ξ 2 ] = ∞, for which we say the distribution of ξ is in the non-normal domain of attraction to the normal law. In Sections 2.2.2, 2.3.2 and 2.4, we then deduce asymptotics of the conditional law of the summands given these large deviations.

Rozovskii's theorem and a few comments

We now state the main part of Theorem 6 in Rozovskii's [START_REF] Rozovskii | Probabilities of large deviations of sums of independent random variables with common distribution function in the domain of attraction of the normal law[END_REF] which is to our knowledge the best result so far for large deviations of random walks in the domain of attraction of the normal law; it is particularly interesting in the non-normal domain of attraction to the normal law. Let us introduce q(x) :=

x 2 σ 2 (x) F (x) .

(2.1)

As noticed above in (1.4), we necessarily have that lim x→∞ q(x) = 0. The idea in [START_REF] Rozovskii | Probabilities of large deviations of sums of independent random variables with common distribution function in the domain of attraction of the normal law[END_REF] is not to put conditions on F (x), but rather on the function q(x). Rozovskii assumes that there is some c > 0 such that x c q(x) is asymptotically equivalent to a non-decreasing function (in Rozovskii's notation, x c q(x) ↑). We prove in Section A.1 that it is equivalent to having that ∃ c > 0 s.t. x c F (x) is asymptotically equivalent to a non-decreasing function.

(2.2)

Theorem 2.1 (Main part of Thm. 6 in [START_REF] Rozovskii | Probabilities of large deviations of sums of independent random variables with common distribution function in the domain of attraction of the normal law[END_REF]). Let α = 2 and (a n ) n≥1 be a normalising sequence as in (1.5), and assume (2.2). Define ω n := ω n (a n ) = a n / | log q(a n )|. Then the following two relations are equivalent:

sup x≥an P(S n -b n ≥ x) Φ(x/a n ) + nF (x) -1 ---→ n→∞ 0 , (2.3) nF (-ω n ) + n ω 2 n σ 2 (ω n ) - a 2 n ω 2 n ---→ n→∞ 0 .
(2.4)

Note that the condition (2.4) gives a condition on the left tail of the distribution; if E[|ξ| 2+δ ] < ∞ for some δ > 0, then (2.4) is verified (with a n = σ √ n).

Some comments on Rozovkii's theorem. We now make several comments: we refer to Appendix A for more discussions (and for the proofs of some of the following claims).

1. The convergences in (2.3) and (2.4) are sensitive to the specific choices of the normalising sequence (a n ) n≥1 . For instance, to make sure that Φ(

x n /a n ) ∼ Φ(x n /a ′ n ) if x n /a n → ∞, it is not enough to have a n ∼ a ′ n : using the standard asymptotics Φ(t) ∼ 1 t √ 2π e -1 2 t 2 as t → ∞, one needs to have ( xn a ′ n ) 2 -( xn an ) 2 = o(1), or equivalently a ′ n = a n (1 + o(a 2 n /x 2 n ))
. Hence, there are cases where (2.3) and (2.4) will hold for some choices of the normalising sequence a n but not for other ones, see Examples A.12 and A.13.

2.

As implied by the above comment, the condition (2.4) is less demanding for some choices of a n than others. In view of (1.5), a natural choice of a n would be to define it by

a 2 n = nσ 2 (a n ) , (2.5) with σ 2 (x) := E[(|ξ -µ| ∧ x) 2 ];
we replaced here σ 2 (•) by σ 2 (•) because σ 2 (y) ∼ σ 2 (y) as y → ∞ (see (A.5)) and because σ 2 (•) is not necessarily continuous. Note that such a definition is always possible since σ 2 is increasing and continuous.

With the choice (2.5), one can verify that a n is a normalising sequence satisfying (1.5) and we show in Proposition A.3 that the condition (2.4) is then equivalent to

σ 2 x | log q(x)| -σ 2 (x) = o σ 2 (x) | log q(x)| , as x → ∞ . (2.6)
In the case where Var(ξ) < ∞, the above condition is strictly weaker than (2.7), see Examples A.12 and A.13. We also provide an example where E[ξ 2 ] = ∞ (hence (2.7) fails) and (2.6) is verified (hence (2.4) holds) for a n defined by (2.5), see Example A.10.

3. As stated in [Roz90, Thm. 6], if Var(ξ) = σ 2 < ∞ and if one chooses a n = σ √ n, the condition (2.4) is equivalent to σ 2 -σ 2 (x) = E[(ξ -µ) 2 1 {|ξ-µ|>x} ] = o 1 log q(x) , as x → ∞ . (2.7)
We give in Examples A.12-A.13 instances where Var(ξ) = σ 2 < ∞ and (2.7) is satisfied (i.e.

(2.3) holds for a n = σ √ n), but Nagaev's criterion fails, i.e. E[|ξ| 2+δ ] = ∞ for all δ > 0.

4. Finally, there exist distributions such that, for any normalising sequence (a n ) n≥1 satisfying (1.5), the condition (2.4) does not hold (hence (2.3) fails), see Example A.14.

It is useful to keep the following example in mind, which generalises Example 1.1 by including the case β = 2.

Example 2.2. Consider a centered random variable ξ, with a right tail that verifies

F (x) := P(ξ > x) ∼ L(x)x -β as x → ∞ , (2.8)
for some β ∈ [2, ∞) and some slowly varying function L(•). Note that compared to Example 1.1, the case β = 2 is included. If Rozovskii's condition (2.4) holds (note that q(x) ∼ x2-β L(x)/σ 2 (x) so q(x) is slowly varying if β = 2), we can check that setting

γ n = x 2 n 2a 2 n -| log q(a n )| - 1 2 (β -1) log | log q(a n )| -log L(a n ) L(a n | log q(a n )|)
(2.9)

(note that x 2 n ∼ 2a 2 n | log q(a n )| if γ n = o(| log q(a n )|), which is the most interesting case), then applying (2.3) we get that if lim n→∞ γ n = γ ∞ ∈ [-∞, ∞] we obtain lim n→∞ P(S n -b n ≥ x n ) nF (x n ) = 1 + c β e -γ∞ , with c β = 1 √ 2π 2 (β-1)/2 .
Hence, the one-big-jump phenomenon (1.7) occurs if and only if lim n→∞ γ n = ∞. Note that this example extends and sharpens [Mog08, Thm. 1.2]; furthermore, assuming only E[ξ] = 0 and (2.8) with β > 1, [Nag82, Thm. 1.1] shows that lim inf n→∞ x n /n > 0 is sufficient for one-big-jump to occur. Let us stress that in Example A.10 in the Appendix, we show that if the left tail verifies F (-x) ∼ cF (x) for some c ≥ 0 (by convention F (-x) = o(F (x)) if c = 0), then Rozovskii's condition (2.4) holds with a n defined by a 2 n = nσ 2 (a n ).

Further comments on the right-tail assumption. Let us introduce the following generalizations of regular variation, called extended and intermediate regular variation, introduced respectively by Matuszewska [START_REF] Matuszewska | On a generalization of regularly increasing functions[END_REF] and Cline [START_REF] Daren | Intermediate regular and Π variation[END_REF]:

(i) A function f is called extended regularly varying if for some real numbers c, d (called upper and lower Matuszewska indices)

λ d ≤ lim inf x→∞ f (λx) f (x) ≤ lim sup x→∞ f (λx) f (x) ≤ λ c for all λ ≥ 1 . (2.10)
In fact, it is known, see [BGT89, Thm. 2.0.7], that the bounds hold locally uniformly in the sense that for any Λ > 1, (1

+ o(1))λ d ≤ f (λx)/f (x) ≤ (1 + o(1))λ c as x → ∞ uniformly in 1 ≤ λ ≤ Λ. (ii) A function f is called intermediate regularly varying if lim λ↓1 lim inf x→∞ f (λx) f (x) = lim λ↓1 lim sup x→∞ f (λx) f (x) = 1 . (2.11)
Let us mention that there is also the notion of O-regular variation (see e.g. [BGT89, Ch. 2.0.2]) which is weaker than intermediate regular variation 2 . We will not use that notion in the paper.

In Appendix A, see Claim A.1, we prove that the condition (2.2) is equivalent to the fact that F is extended regularly varying. In the following we will mostly work assuming the slightly weaker condition that F is intermediate regularly varying. We stress that both conditions imply that the law of ξ is subexponential, i.e. for any y ∈ R we have lim x→∞ F (x + y)/F (x) = 1. We refer to Section A.1 for further comments.

After a careful review of [START_REF] Rozovskii | Probabilities of large deviations of sums of independent random variables with common distribution function in the domain of attraction of the normal law[END_REF], we believe that Rozovskii's Theorem 2.1 still holds assuming intermediate regular variation, in place of extended regular variation. However [START_REF] Rozovskii | Probabilities of large deviations of sums of independent random variables with common distribution function in the domain of attraction of the normal law[END_REF] is very intricate so we are not confident enough to make a definite claim.

Example 2.3. A classical example where F is intermediate regularly varying but not regularly varying at ∞ is the following:

F (x) ∼ x -γ+sin(log log x) ,
where γ has to be larger than √ 2 in order for the r.h.s. to be non-increasing in x (computing the derivative, a factor sin(log log x) + cos(log log x) -γ appears). Note that the upper and lower Matuszewska indices are then -γ + 1 and -γ -1 respectively. Let us stress that for F to be in the domain of attraction of a normal law, one also needs to have γ ≥ 3.

First set of results: Large deviations and conditional laws

We now collect a few results on large deviation probabilities for S n . Our main assumption, aside from the fact that ξ is in the domain of attraction of a normal law, will be that F is intermediate regularly varying, i.e. (2.11) (if we want to use Rosovskii's theorem, we will assume the stronger condition that F is extended regularly varying, i.e. (2.10)). In all the following, we assume that α = 2 and we let (a n ) n≥1 be a normalising sequence, see (1.5).

Large deviations

Let us consider a function r : (0, ∞) → R + that satisfies the asymptotic relation r(t) σ 2 (r(t)) ∼ t as t → ∞. Such a function exists because σ 2 (x) is slowly varying at infinity and thus we have lim t→∞ r(t) = ∞.

Then for any positive sequence (x n ) n≥1 we define

r n := r n x n .
(2.12)

In fact, we will mostly focus on the case where lim n→∞ xn n = 0, in which case r n verifies

lim n→∞ r n = ∞ ; r n σ 2 (r n ) ∼ n x n .
(2.13)

In order to better explain the estimates of P(S n -b n ≥ x n ), we make the following decomposition:

P(S n -b n ≥ x n ) = P(S n -b n ≥ x n , M n ≤ r n ) + P(S n -b n ≥ x n , M n > r n ). (2.14)
Analysis of the first term in the r.h.s. of (2.14). Let us introduce, for u ≥ 0, the tilted (and truncated) distribution

P u = P (rn) u defined by dP u dP (x) = 1 M (u) e u(x-µ) 1 (-∞,rn] (x) with M (u) = M rn (u) := E e u(ξ-µ) 1 {ξ≤rn} . (2.15)
Define also, for u ≥ 0,

m(u) = M ′ (u) M (u) = E u [ξ -µ] (2.16)
and its inverse (u → m(u) is increasing: we can show that m ′ (u) > 0 for any u ≥ 0)

λ(t) := m -1 (t) (2.17) for t ≥ E 0 [ξ -µ] (note that E 0 [ξ -µ] < 0), so that E λ(t) [ξ] = µ + t. Finally, let H(t) = H rn (t) := -log M (λ(t)) + tλ(t) , (2.18)
which is the relative entropy of P λ(t) w.r.t. to P. Note that we allow a small abuse of notation here since P u will later refer both to the law of a single variable and to the law of n i.i.d. copies of that variable. With this notation we can state the following result.

Proposition 2.4. Let α = 2 and (a n ) n≥1 be a normalising sequence as in (1.5). Let (x n ) n≥1 be a sequence such that x n ≥ a n and lim n→∞ xn n = 0, and let r n be as in (2.12). Then

P(S n -b n ≥ x n , M n ≤ r n ) ∼ 1 √ 2π a n x n σ 2 (r n ) σ 2 (a n ) e -nH( xn n ) as n → ∞ . (2.19) Remark 2.5. Let us stress that if a n ≤ x n ≤ Ca n | log q(a n )| for some constant C > 0,
then assuming that F is intermediate regularly varying (see (2.11)) we get that σ 2 (r n ) ∼ σ 2 (a n ), see Remark A.9 in the Appendix. Note also that the case where

x n = O(a n ) is in the regime of the usual central limit theorem (with the extra constraint M n ≤ r n = O(a n )).
We also provide the following lemma that estimates the relative entropy nH( xn n ).

Lemma 2.6. Let α = 2 and (a n ) n≥1 be a normalising sequence as in (1.5). Let (x n ) n≥1 be a sequence such that x n ≥ a n and lim n→∞ xn n = 0, and let r n be as in (2.12). Then we have

nH x n n = (1 + o(1)) 1 2 x 2 n nσ 2 (r n ) = (1 + o(1)) 1 2 x 2 n a 2 n σ 2 (a n ) σ 2 (r n ) . Remark 2.7. Note that if xn an = O(1), then nH( xn n ) = (1+o(1)) x 2 n 2a 2 n = x 2 n 2a 2 n +o(1) (also using that σ 2 (r n ) ∼ σ 2 (a n ), see Remark 2.5).
In view of the above lemma, Rozovskii's theorem actually tells that, under (2.2), condition (2.4) is a criterion to ensure that nH

( xn n ) = x 2 n 2a 2 n + o(1) uniformly for a n ≤ x n ≤ Ca n | log q(a n )|; the r.h.s. of (2.19) is then asymptotically equivalent to Φ( xn an ). Note that the threshold a n | log q(a n )| appears when comparing Φ( xn an ) with nF (x n ): if x n ≥ Ca n | log q(a n )| with C > √ 2 then (2.19) becomes negligible compared to nF (x n ), see Remark A.6 in the Appendix.
Analysis of the second term in the r.h.s. of (2.14). For the remaining term in (2.14), we have the following result.

Proposition 2.8. Let α = 2 and (a n ) n≥1 be a normalising sequence as in (1.5). Assume that F is intermediate regularly varying, see (2.11). If lim n→∞ xn an = ∞ and lim n→∞ xn n = 0, then letting r n be as in (2.12), we have

P(S n -b n ≥ x n , M n > r n ) ∼ P(S n -b n ≥ x n , M n ≥ x n ) , or equivalently P(S n -b n ≥ x n ) = P(S n -b n ≥ x n , M n ≤ r n ) + (1 + o(1))nF (x n ) .
(2.20)

If lim inf n→∞ xn n > 0, then P(S n -b n ≥ x n ) ∼ P(S n -b n ≥ x n , M n ≥ x n ) ∼ nF (x n ).
The statement with lim inf n→∞ xn n > 0 is standard, see e.g. [START_REF] Denisov | Large deviations for random walks under subexponentiality: the big-jump domain[END_REF]. We include it in our proof for the sake of completeness. Let us also mention that if F is only O-regularly varying, we could replace the 1 + o(1) in (2.20) by a O(1); we do not develop this further here.

Remark 2.9. Denisov, Dieker and Shneer [START_REF] Denisov | Large deviations for random walks under subexponentiality: the big-jump domain[END_REF] give conditions for the appearance of the one-big-jump phenomenon in the general context of subexponential distributions. Since we work here under much more restricted assumptions, we are able to to give sharper conditions for the minimal size of x n to be in the one-big-jump regime for specific examples: [START_REF] Denisov | Large deviations for random walks under subexponentiality: the big-jump domain[END_REF]

√ | log q(an)| P(S n > y) nF (y) -1 = 0, if c > √ 2, > 0, if 0 ≤ c < √ 2.
Using the nomenclature from [START_REF] Denisov | Large deviations for random walks under subexponentiality: the big-jump domain[END_REF], we see that in this case if we set x n = ca n | log q(a n )| with c > √ 2 and define r n via (2.13), then (r n ) is a "truncation sequence" and (x n ) is a "(r n )-small-steps sequence" (we can use √ n as the "natural-scale sequence" and any sequence (I n ) with √ n/I n → 0 as "insensitivity sequence").

If in this situation we impose that F is in fact regularly varying, we can read off the precise behaviour from Example 2.2: A given sequence (x n ) is a small-steps sequence in the sense of [START_REF] Denisov | Large deviations for random walks under subexponentiality: the big-jump domain[END_REF] if and only if the sequence (γ n ) defined in (2.9) diverges. An analogous situation occurs in the local case, see Example 2.17.

Conditional laws on large deviation events

Using Propositions 2.4 and 2.8 (and their proofs), we can make the one-big-jump phenomenon more precise: The intuition behind (1.7) is of course that only one random variable absorbs the large deviation, without effectively changing the distribution of the n -1 others. The following results make the fact that the remaining random variables are "left untouched" explicit by showing that their distribution is close to that of n -1 i.i.d. random variables with the same law as ξ.

For y = (y 1 , . . . , y n ) ∈ R n , we define R(y) ∈ R n-1 , the vector obtained by removing the i-th coordinate from y, where |y i | = max 1≤j≤n |y j | =: M n (y). In case of a tie for the maximum, we take away the variable with the index min{i : y i = M n (y)}. In fact, other choices are possible and will lead to the same limit behaviour as in Theorem 2.10 and in Proposition 4.1 below. In the following, we write L (X) to denote the law of a random variable (or a vector of random variables) X.

Theorem 2.10 (One-big-jump phenomenon). Let α = 2 and (a n ) n≥1 be a normalising sequence, see (1.5). Assume that F is intermediate regularly varying, see (2.11). If lim n→∞ xn an = ∞ holds, and then letting r n be as in (2.12), we have

lim n→∞ d TV L R(ξ 1 , . . . , ξ n ) S n -b n ≥ x n , M n > r n , L (ξ) ⊗(n-1) = 0, (2.21) lim n→∞ d TV L R(ξ 1 , . . . , ξ n ) S n -b n ≥ x n , M n ≤ r n , L (ξ) ⊗(n-1) = 1, (2.22)
where d TV denotes the total variation distance. As a consequence, if in addition (x n ) satisfies lim n→∞ nF (xn) For the value of s in the context of Example 2.2, recalling the definition (2.9) of γ n , we have s =

P(Sn≥xn) = s ∈ [0, 1], then we have lim n→∞ d TV L R(ξ 1 , . . . , ξ n ) S n -b n ≥ x n , L (ξ) ⊗(n-1) = 1 -s . ( 2
1 1+c β e -γ∞ ∈ [0, 1], with γ ∞ = lim n→∞ γ n ∈ [-∞, ∞] and c β as in Example 2.2.
Remark 2.11. 1. The situation is in fact much simpler if unlike in Theorem 2.10, one only conditions on M n being atypically large. We address this briefly in Section 4.1 later.

If lim

n→∞ xn an ∈ [-∞, ∞), then it is natural to expect that lim n→∞ d TV L R(ξ 1 , . . . , ξ n ) S n -b n ≥ x n , L (ξ) ⊗(n-1) > 0.
This can be proved along the lines of the argument for the necessity in Corollary 3.2, which will be proved in Appendix B.1. We omit the details for the proof of the above statement.

In the course of the proofs of Propositions 2.4 and 2.8, we also obtain information on the overshoot S n -b n -x n , conditioned on the large deviation event {S n -b n ≥ x n }.

Corollary 2.12. Let α = 2 and (a n ) n≥1 be a normalising sequence, see (1.5), and assume that lim n→∞ xn an = ∞. Then we have 1. If lim n→∞ xn n = 0, then conditionally on having no big-jump (M n ≤ r n ),

L S n -b n -x n x n /r n S n -b n ≥ x n , M n ≤ r n w ---→ n→∞ Exp (
P(S n -b n ≥ x ′ n | S n -b n ≥ x n , M n > r n ) ∼ F (x ′ n ) F (x n ) . (2.25)
In particular, if F varies regularly with index -β for some β ≥ 2, the last part gives that, for y ∈ [1, ∞)

lim n→∞ P S n -b n x n ≥ y S n -b n ≥ x n , M n > r n = y -β , (2.26)
i.e., conditioned on {S n -b n ≥ x n , M n > r n }, the overshoot behaves asymptotically like x n times a Pareto random variable. In particular, in the regime where s = lim n→∞ nF (xn) P(Sn-bn≥xn) ∈ (0, 1), the conditional law of the overshoot will be a non-trivial mixture of a scaled exponential and of a (differently) scaled Pareto distribution.

Second set of results: Local large deviations and conditional laws

In this section, we study local versions of the large deviations, namely we obtain estimates on the probabilities of the type P(S n -b n ∈ [x n , x n + ∆)) in the regime lim n→∞ xn an = ∞. To simplify the statements, we will assume that ξ is integer valued and that (S n ) n≥1 is aperiodic (the latter is true if F is intermediate regularly varying). The case that ξ has support in a + hZ for some h > 0 and a ∈ R is completely analogous, just notationally a little more cumbersome.

To obtain sharp results, we need some local condition on the distribution of ξ. Our main assumption is that

P(ξ = x)
P(ξ = x) ∼ βL(x)x -(1+β) as x → ∞ .
(2.28)

In particular, we have that F (x) ∼ L(x)x -β . Doney [START_REF] Doney | A local limit theorem for moderate deviations[END_REF] proves that if in addition to (2.28) we have E[|ξ| 2+δ ] < ∞ for some δ > 0 (implying β > 2), then analogously to Nagaev's result for the integral case [Nag79, Thm. 1.9], see (1.9),

P(S n -⌊b n ⌋ = x n ) = (1 + o(1)) 1 σ √ n g x n σ √ n + (1 + o(1))nP(ξ = x n ) , (2.29)
where g(t) = 1 √ 2π e -t 2 2 is the standard normal density and where σ 2 = Var(ξ). Let us mention that [Mog08, Thm. 2.2] gives the same result, assuming β > 2 in (2.28) and with the moment condition E[ξ 2 1 {|ξ|>x} ] = o(1/ log x) as x → ∞, analogously to (2.7).

Example 2.14. Continuing the discussion in Example 2.13, assume that (2.28) holds with β > 2 and E[|ξ| 2+δ ] < ∞ for some δ > 0. Then since (2.29) holds, it is a standard calculation to obtain that letting

γn := x 2 n 2σ 2 n - β 2 -1 log n - 1 2 (β + 1) log log n + log L( n log n) with σ 2 = Var(ξ), then if lim n→∞ γn = γ∞ ∈ [-∞, ∞], we obtain lim n→∞ P(S n -⌊b n ⌋ = x n ) nP(ξ = x n ) = 1 + cβ,σ e -γ∞ , with cβ,σ = σ β √ 2π β -1 (β -2) (β+1)/2 .
Hence the one-big-jump phenomenon occurs if and only if lim n→∞ γn = ∞. Let us stress that, compared with Example 1.1, the definition of γn differs from that of γ n in the constant in front of log log n. In particular, we might have γ n → ∞ while γn → -∞. In other words, there is a regime where the one-big-jump phenomenon occurs for the large deviations but not for the local large deviations.

Our goal is to extend the asymptotics (2.29) of [START_REF] Doney | A large deviation local limit theorem[END_REF][START_REF] Anatolii | An integro-local theorem that is applicable on the whole half-axis for sums of random variables with regularly varying distributions[END_REF] to include non-normal domains of attraction to the normal law: in particular, in (2.28) we include the case β = 2.

Local large deviations

As for the (integral) large deviation case, we split the local large deviation probability according to the threshold r n defined in (2.12):

P(S n -⌊b n ⌋ = x n ) = P(S n -⌊b n ⌋ = x n , M n ≤ r n ) + P(S n -⌊b n ⌋ = x n , M n > r n ) .
We analyse the two terms through two separate results respectively. Proposition 2.15. Let α = 2 and let (a n ) n≥1 be a normalising sequence as in (1.5); assume also that (S n ) n≥0 is aperiodic. If (x n ) n≥1 is a sequence such that x n ≥ a n and lim n→∞ xn n = 0, then letting r n be as in (2.12), we have as n → ∞ = 0, then letting r n be as in (2.12), we have for any ε > 0

P(S n -⌊b n ⌋ = x n , M n ≤ r n ) ∼ 1 2πnσ 2 (r n ) e -nH( xn n ) . ( 2
P(S n -⌊b n ⌋ = x n , M n > r n ) ∼ P (S n -⌊b n ⌋ = x n , |M n -x n | ≤ εx n ) ,
(2.31)

and also

P(S n -⌊b n ⌋ = x n ) = P(S n -⌊b n ⌋ = x n , M n ≤ r n ) + (1 + o(1))nP(ξ = x n ) . If lim inf n→∞ xn n > 0, then we have P(S n -⌊b n ⌋ = x n ) ∼ P (S n -⌊b n ⌋ = x n , |M n -x n | ≤ εx n ) ∼ nP(ξ = x n ) .
The statement with lim inf n→∞ xn n > 0 is standard, see e.g. [AL09, Prop. 1]. We include it in our proof for the sake of completeness.

Let us mention that Remark 2.7 also applies here in the local case. If additionally to the assumptions in Proposition 2.16 condition (2.4) holds, then thanks to Lemma 2.6 we have nH

xn n = x 2 n 2a 2 n + o(1) uniformly for a n ≤ x n ≤ Ca n | log q(a n )|. Therefore, if lim n→∞ xn an = ∞ and x n ≤ Ca n | log q(a n )|,
thanks to Propositions 2.15 and 2.16 we obtain

P(S n -⌊b n ⌋ = x n ) = (1 + o(1)) 1 a n g x n a n + (1 + o(1))nP(ξ = x n ) , (2.32)
where we recall g(•) is the standard normal density (and we have used that nσ 2 (r n ) ∼ nσ 2 (a n ) ∼ a n in (2.30), see Remark 2.5). Example 2.17. Let ξ be an integer-valued random variable that satisfies (2.28). We also assume that Rozovskii's condition (2.4) holds, see Example A.10 for a simple sufficient condition. Note that compared to Example 2.14, we include the case

β = 2. If we set γn = x 2 n 2a 2 n -| log q(a n )| - 1 2 (β + 1) log | log q(a n )| -log L(a n ) L(a n | log q(a n )|) , (2.33) then if lim n→∞ γn = γ∞ ∈ [-∞, ∞], thanks to (2.32) we obtain that lim n→∞ P(S n -⌊b n ⌋ = x n ) nP(ξ = x n ) = 1 + cβ e -γ∞ , with cβ = 1 √ 2π β -1 2 (β+1)/2 .
Hence the one-big-jump phenomenon occurs if and only if lim n→∞ γn = ∞. We stress again that, compared with Example 2.2, the definition (2.33) of γn differs from that (2.9) of γ n in the prefactor of log | log q(a n )|; in particular we might have γ n → ∞ while γn → -∞.

Remark 2.18. Let us mention that even without the local assumption that P(ξ = x) is (intermediate) regularly varying, one can still obtain useful estimates on the local large deviation probabilities. Assuming that α = 2 and letting (a n ) n≥1 be a normalising sequence as in (1.5), [MT22, Thm. 1.1] gives the following bound (with optimal decay rate): uniformly for

x n ≥ a n P(S n -⌊b n ⌋ = x n ) ≤ C a n × nσ 2 (x n ) x 2 n .
(2.34)

If one has additionally that F (x) ≤ L(x)x -β for some β ≥ 2 and some slowly varying function L(•), then [Ber19b, Thm. 2.1] gives that there are positive constants c 1 , C 1 such that uniformly in x n ≥ a n ,

P(S n -⌊b n ⌋ = x n ) ≤ C 1 a n e -c 1 x 2 n /a 2 n + nL(x n )x -β n .
This is obviously not as sharp as (2.29) (in particular the constant c 1 in the exponential is not optimal) but it is sharper than (2.34) when x n ≫ a n . The above display requires very little assumption and can be useful in several situations.

Conditional laws on local large deviation events

Recall the definition of the map y → R(y) for y = (y 1 , . . . , y n ) ∈ R n from Section 2.2.2.

Theorem 2.19 (Local one-big-jump phenomenon). Let α = 2, let (a n ) n≥1 be a normalising sequence as in (1.5), and assume that P(ξ = x) is intermediate regularly varying (see (2.11)).

Let (x n ) n≥1 be a sequence such that lim n→∞ xn an = ∞ and let r n be as defined in (2.12). Then we have

lim n→∞ d TV L R(ξ 1 , . . . , ξ n ) S n -⌊b n ⌋ = x n , M n > r n , L (ξ) ⊗(n-1) = 0 , lim n→∞ d TV L R(ξ 1 , . . . , ξ n ) S n -⌊b n ⌋ = x n , M n ≤ r n , L (ξ) ⊗(n-1) = 1 .
As a consequence, if in addition (2.27) holds and (x n ) satisfies lim n→∞ nP(ξ=xn)

P(Sn-⌊bn⌋=xn) = s ∈ [0, 1], then we have lim n→∞ d TV L R(ξ 1 , . . . , ξ n ) S n -⌊b n ⌋ = x n , L (ξ) ⊗(n-1) = 1 -s. (2.35)
For the value of s in the context of Example 2.17, recalling the definition (2.33) of γn , we have s =

1 1+c β e -γ∞ ∈ [0, 1], with γ∞ = lim n→∞ γn ∈ [-∞, ∞]
and cβ as in Example 2.17.

Remark 2.20. Similarly to Remark 2.11, when we have lim n→∞

xn an ∈ [-∞, ∞)
, it is natural to expect the following (the proof is omitted)

lim n→∞ d TV L R(ξ 1 , . . . , ξ n ) S n -⌊b n ⌋ = x n , L (ξ) ⊗(n-1) > 0.
Remark 2.21. In view of the proof of Proposition 2.15, we could also prove that

lim n→∞ d TV L ξ 1 , . . . , ξ n S n -⌊b n ⌋ = x n , M n ≤ r n , P n,xn = 0 , (2.36) where P n,xn := L (( ξ(n) 1 , . . . , ξ(n) n ) | n i=1 ξ(n) n = x n ), with ( ξ(n) i ) 1≤i≤n i.i.d. random variables with law P λ( xn n ) defined in (2.15) (recall also the definition of λ(t) in the line that follows, so ξ(n) i have mean xn n ).
In other words, conditioned on having no big-jump (M n ≤ r n ), the random variables behave as tilted random variables with individual mean xn n , conditioned on having their sum equal to their collective mean x n . The statement is somehow cumbersome and not technically useful, so we decided not to provide a proof for it.

We mention that one could use this observation to obtain an analogous result in the integrated case L (ξ 1 , . . . , ξ n | S n -⌊b n ⌋ ≥ x n , M n ≤ r n ): one would need to integrate the conditioning {S n -⌊b n ⌋ = y, M n ≤ r n } for y ≥ x n , hence obtaining a mixture of ( P n,y ) y≥xn for integers y ≥ x n . But this goes beyond the scope of our paper.

Conditional law of the maximum on a large deviation event

To complement the previous results, we study the law of the maximum summand conditioned on the sum being large. We will prove only the first corollary as the second one can be proved following similar arguments.

Corollary 2.22. Let α = 2, let (a n ) n≥1 be a normalising sequence as in (1.5) and assume that F is intermediate regularly varying (see

(2.11)). If lim n→∞ xn an = ∞ and if lim n→∞ nF (xn) P(Sn-bn≥xn) = s ∈ [0, 1], then we have L M n S n -b n S n -b n ≥ x n w ---→ n→∞ (1 -s)δ 0 + sδ 1 .
Corollary

P(Sn-⌊bn⌋=xn) = s ∈ [0, 1], we have L M n x n S n -⌊b n ⌋ = x n w ---→ n→∞ (1 -s)δ 0 + sδ 1 .
From the above two corollaries, a natural question is to understand the distribution of (rescaled versions of) M n or M n -(S n -b n ) conditionally on a large deviation event (either local or integral). We discuss this in Section 4.3 later.

Organisation of the rest of the paper

Let us give a brief overview on how the rest of the paper is organised:

• In Section 3, we study the summands in the domain of attraction of a stable law with index α < 2.

• In Section 4, we discuss some applications and possible extensions of our results.

• In Section 5, we prove the main large deviation results of Section 2.2 above. After some preliminary estimates, we prove Proposition 2.4 and then Proposition 2.8. Finally, we give proofs for Theorem 2.10, Corollary 2.12 and Corollary 2.22.

• In Section 6, we prove the local large deviation results of Section 2.3. We start with the proof of Proposition 2.15, then we prove Proposition 2.16 and finally Theorem 2.19.

• Section 7 is dedicated to the proof of Proposition 4.1, which is carried out after having constructed the regular conditional distribution of R(ξ 1 , . . . , ξ n ) given M n .

• Finally, the appendix is dedicated to some technical results and comments. Appendix A contains several comments, examples and proofs of claims regarding Rozovkii's Theorem 2.1. Appendix B deals with the case of random variables in the domain of attraction of an α-stable law with α ∈ (0, 2), proving in particular Corollaries 3.2 and 3.5.

3 The case of the domain of attraction of an α-stable law with α ∈ (0, 2)

The one-big-jump phenomenon is well understood in the case where ξ is in the domain of attraction of an α-stable law with α ∈ (0, 2): in a nutshell, one has a "big-jump" whenever lim n→∞ xn an = ∞, see [CH89; Nag79] for the case α ∈ (0, 1) ∪ (1, 2) and [START_REF] Berger | Notes on random walks in the Cauchy domain of attraction[END_REF] for the case α = 1. In order to put our results in perspective we recall here some results for large and local large deviations and give their consequences in terms of conditional laws.

Large deviations and conditional laws

Assume that α ∈ (0, 2) and recall the definitions (1.5)-(1.6) of the normalising and centering sequences (a n ) n≥1 , (b n ) n≥1 . Then Theorem 2.1 in [START_REF] Berger | Notes on random walks in the Cauchy domain of attraction[END_REF] collects large deviation results in that setting.

Theorem 3.1 (Thm. 2.1 in [START_REF] Berger | Notes on random walks in the Cauchy domain of attraction[END_REF]). Assume that α ∈ (0, 2), let (a n ) n≥1 , (b n ) n≥1 be the sequences in (1.1), and recall (1.2). If lim n→∞ xn an = ∞, then we have, as n → ∞,

P(S n -b n ≥ x n ) ∼ npL(x n )x -α n , P(S n -b n ≤ -x n ) ∼ nqL(x n )x -α
n . If p = 0 or q = 0, one interprets the corresponding term on the r.h.s as o(nL(x)x -α ).

As a corollary, we obtain a result on the law of ξ 1 , . . . , ξ n conditionally on the large deviation event {S n -b n ≥ x n }.

Corollary 3.2 (One-big-jump phenomenon). Assume α ∈ (0, 2), let (a n ) n≥1 , (b n ) n≥1 be the sequences in (1.1) and suppose that p > 0 in (1.2). If lim n→∞ xn an = ∞, then as n → ∞ P(S n -b n ≥ x n ) ∼ P(S n -b n ≥ x n , M n ≥ x n ) ∼ P(M n ≥ x n ). (3.1)
Moreover the condition lim n→∞ xn an = ∞ is necessary and sufficient for

lim n→∞ d TV L R(ξ 1 , . . . , ξ n ) S n -b n ≥ x n , L (ξ) ⊗(n-1) = 0. (3.2)
In particular, if lim n→∞ xn an = ∞, we have

L S n -b n -M n a n S n -b n ≥ x n w ---→ n→∞ L (S α ), (3.3)
where w -→ denotes weak convergence and S α is the α-stable random variable appearing in (1.1).

Remark 3.3. When p = 0, Theorem 3.1 only gives that

P(S n -b n ≥ x n ) = o(nL(x n )x -α n ).
Thus, the precise asymptotic behaviour of the large deviation probability P(S n -b n ≥ x n ) remains in general open; note that (S n -b n )/a n converges in law to a spectrally negative α-stable random variable. We mention [DDS08, Thms. 9.2 and 9.3] and also [START_REF] Aleksandr | Large deviations probabilities for random walks in the absence of finite expectations of jumps[END_REF] for sufficient conditions to have a one-big-jump phenomenon, but we are not aware of a necessary and sufficient condition in this case.

Local large deviations and conditional laws

To simplify the statements, we assume that ξ is integer valued. We are interested in estimating the local large deviation probabilities P(S n -⌊b n ⌋ = x n ) as n → ∞, if (x n ) n≥1 is a sequence of integers such that lim n→∞ xn an = ∞. For this type of results, we need an extra condition on the local tail behaviour of the distribution of ξ: in addition to (1.2), we assume that there is some α ∈ (0, 2) and some slowly varying function L(•) such that

P(ξ = x) ∼ pαL(x)x -(1+α) , as x → ∞ , (3.4)
with p ≥ 0; note that this implies the first half of (1.2). If p = 0, one interprets it as o(nL(x)x -(α+1) ). We now recall the result obtained in [START_REF] Berger | Notes on random walks in the Cauchy domain of attraction[END_REF] (see also [START_REF] Doney | One-sided local large deviation and renewal theorems in the case of infinite mean[END_REF] for the case α ∈ (0, 1)).

Theorem 3.4 (Thm. 2.4 in [START_REF] Berger | Notes on random walks in the Cauchy domain of attraction[END_REF]). Assume that (1.2) holds for some α ∈ (0, 2) and that additionally one has (3.4); let (a n ) n≥1 , (b n ) n≥1 be the sequences in (1.1). Then, if lim n→∞ xn an = ∞, we have

P(S n -⌊b n ⌋ = x n ) ∼ npαL(x n )x -(1+α) n .
(3.5)

Corollary 3.5 (Local one-big-jump phenomenon). Assume α ∈ (0, 2), let (a n ) n≥1 , (b n ) n≥1
be the sequences in (1.1) and assume that (3.4) holds for some p > 0. Then the condition lim n→∞ xn an = ∞ is necessary and sufficient for

lim n→∞ d TV L R(ξ 1 , . . . , ξ n ) S n -⌊b n ⌋ = x n , L (ξ) ⊗(n-1) = 0. (3.6)
In particular, under lim n→∞ xn an = ∞,

L x n -M n a n S n -⌊b n ⌋ = x n w ---→ n→∞ L (S α ). ( 3
P(S n -⌊b n ⌋ = x n ) ≤ C 0 a n nP(|ξ| ≥ x n ) .

Further comments, applications and open questions

The behaviour of summands in a sum of i.i.d. random variables is of course a ubiquitous and classical theme in probability theory: the phenomenon of one big-jump being responsible for atypically large values of the sum features in many applications. Let us mention here random walks and renewal processes (see, e.g. [Ber19b; Ber19a] and the references there), population models in regimes with asymptotically infinite offspring variance (where reproduction proceeds in two steps, with each of N individuals first generating a random number ξ i of juveniles, from which then N are sampled to form the next generation, as e.g. in [Sch03; BLS18; BY23]), the zero-range process on finite graphs subject to certain homogeneity conditions (since then the stationary law of the occupation numbers is i.i.d. conditioned on the sum, one big-jump corresponds to condensation, see Section 4.2 below), critical Galton-Watson trees with heavy-tailed offspring distribution (see e.g. [START_REF] Kortchemski | Limit theorems for conditioned non-generic Galton-Watson trees[END_REF] and [START_REF] Kortchemski | Condensation in critical Cauchy Bienaymé-Galton-Watson trees[END_REF], where conditioned on being large, a node with macroscopic degree may emerge) and their applications in the study of random planar maps (e.g. [START_REF] Kortchemski | The boundary of random planar maps via looptrees[END_REF]), ruin problems in insurance mathematics (e.g. [START_REF] Beck | A new class of large claim size distributions: Definition, properties, and ruin theory[END_REF]). Note also that for i.i.d. sums with exponential tails, if in the so-called borderline case the appropriately tilted laws are sufficiently heavy-tailed, even richer behaviour than one big jump may appear when conditioning on reaching an atypically large value, see [START_REF] Sergey | On the limit law of a random walk conditioned to reach a high level[END_REF] for instance.

Conditional laws when M n is large

In Section 2.2.2, we considered the conditional joint law of the n -1 smallest variables conditioned on a large deviation of the sum. Let us briefly revisit this for the case when one instead conditions on an atypically large value of the maximum. Recall the definition of the map y → R(y) for y = (y 1 , . . . , y n ) ∈ R n from Section 2.2.2.

In the following, we write L (X) to denote the law of a random variable (or a vector of random variables) X. Let us mention that a regular conditional distribution of

R(ξ 1 , . . . , ξ n ) given M n exists, that is L R(ξ 1 , . . . , ξ n ) M n = x (4.1)
is well defined for any x in the support of ξ. In fact, we give a concrete construction for the regular conditional distribution, see Section 7.

The following result is very natural when considering a large deviation event that involves only the maximum M n . We were unable to find a reference for it, so we will prove it in Section 7.2. Proposition 4.1. Let (x n ) n≥1 be a sequence satisfying lim n→∞ nF (x n ) = 0, with F (x n ) > 0 for all n. Then

lim n→∞ d TV L R(ξ 1 , . . . , ξ n ) M n ≥ x n , L (ξ) ⊗(n-1) = 0, (4.2) and lim n→∞ d TV L R(ξ 1 , . . . , ξ n ) M n = x n , L (ξ) ⊗(n-1) = 0 . (4.3)
Note that Proposition 4.1 requires no structural conditions on the distribution of the ξ's.

Application to the zero-range process

As an example of application, we discuss in detail how Corollary 2.23 can be applied to the zero-range process. We provide a short introduction to the zero-range process, using the terminology in [START_REF] Armendáriz | Zero-range condensation at criticality[END_REF] (up to minor differences) and rephrasing certain results in our notation. Our goal is to state (part of) Theorem 2.1 in [START_REF] Armendáriz | Zero-range condensation at criticality[END_REF] and explain how our Corollary 2.23 extends it.

Consider a finite set Λ L of L sites. Each site can host any number of (indistinguishable) particles. Informally, the zero-range process is a continuous-time Markov chain with the following dynamics: a site x ∈ Λ L loses a particle at rate g(η x ), where η x is the number of particles at site x and g : N 0 → [0, ∞) is a function such that g(k) = 0 if and only if k = 0. That particle jumps to site y ∈ Λ L with probability p(x, y), where p(•, •) is a transition kernel on Λ L × Λ L (which is assumed to induce an irreducible Markov chain on Λ L , with a spatially homogeneous invariant measure). To proceed, we need some more notation as follows:

• A particle configuration is denoted η = (η x : x ∈ Λ L );

• for a configuration η with η x > 0, define the configuration η x,y = (η x,y z ) z∈Λ L by η x,y x = η x -1, η x,y y = η y + 1 and η x,y z = η z for all z = x, y. Then, the zero-range process is described by the following generator on the space of configurations:

Lf (η) = x,y∈Λ L g(η x )p(x, y) (f (η x,y ) -f (η)) ,
which corresponds to the description given above. Note that the total number of particles is preserved, and we denote it by N . Setting w(n

) := n k=1 1 g(k)
, it is known that there is a family of invariant measures, that are product measures indexed by a parameter ϕ ≥ 0 (called fugacity), that are defined by

ν ϕ (η) = 1 z(ϕ) Λ L x∈Λ L w(η x )ϕ ηx , with z(ϕ) = ∞ n=0 w(n)ϕ n ,
as soon as ϕ is such that the normalising constant verifies z(ϕ) < ∞. Then, one can easily observe that since the number of particles S L (η) := x∈Λ L η x is invariant under the microscopic dynamics, the invariant measure on the set of configurations {η :

S L (η) = N } is given by µ N,L (•) = ν ϕ • S L (η) = N . (4.4)
The measure µ N,L is in fact independent of the choice of ϕ. Then, a natural question is whether the convergence of the measures µ N,L holds in the so-called thermodynamic limit, i.e. taking L, N → ∞ with N L → ρ for some ρ > 0. Since [START_REF] Evans | Phase transitions in one-dimensional nonequilibrium systems[END_REF], a lot of interest has been put on the case where the jump rate g(n) decays with the number of particles. A natural assumption is that g(k) = 1 + b k + ε k k for some b > 1 and some vanishing sequence (ε k ) k≥0 . Note that this implies that3

w(n) = n k=1 1 g(k) ∼ L(n)n -b as n → ∞ , (4.5)
for some slowly varying function

L(n) ∼ c ′ exp( n k=1 ε k k ). In fact, in a large part of the literature the choice g(k) = 1 + b k or g(k) = (k/(k -1)) b is made, so that w(n) is explicit with w(n) ∼ cn -b , see [Eva00; AL09; Xu20]. Let us stress that when ϕ = 1 we have ν 1 (η x = n) = w(n) n≥1 w(n) ∼ cL(n)n -b ,
which therefore corresponds to the assumption (2.28) with b = β + 1 (note also that η x is a non-negative random variable). We can then define the critical density

ρ c := E ν 1 [η x ] ∈ (0, ∞].
Then, if L, N → ∞ with N L → ρ for some ρ > 0, we have that (see [START_REF] Jeon | Size of the largest cluster under zerorange invariant measures[END_REF]):

• if ρ < ρ c , the particle distribution µ N,L converges to the limit stationary product measure ν ϕ with ϕ < 1 determined by E νϕ [η x ] = ρ, in the sense of finite-dimensional marginals;

• if ρ ≥ ρ c , then we obtain the same result as above except with ϕ = 1. Additionally, a condensation phase transition occurs: if ρ > ρ c , then there exists a site containing a positive fraction of the particles, whereas if ρ < ρ c no site contains a nonzero fraction of particles, see [START_REF] Großkinsky | Condensation in the zero range process: stationary and dynamical properties[END_REF]. The appearance of a condensation phenomenon can be understood when considering (4.4): taking ϕ = 1 so that ν 1 (η x = n) = w(n) n≥1 w(n) , we obtain that the law µ N,L corresponds to the law of i.i.d. random variables (with a heavytail distribution) conditioned by their sum being equal to

N ∼ ρL ≫ ρ c L = E ν 1 [η 1 ]L.
In particular, a quantity of interest is M L (η) = max{η x : x ∈ Λ L }, which is the largest number of particles that a site contains; we refer to [START_REF] Armendáriz | Thermodynamic limit for the invariant measures in supercritical zero range processes[END_REF] for some asymptotic results on M L in the case where N/L → ρ > ρ c .

In [START_REF] Armendáriz | Zero-range condensation at criticality[END_REF], the authors focus on the critical case where 1 ≪ N -ρ c L = o(L) (we have N/L → ρ c ), with a parameter b > 3 in (4.5), so that in particular ν 1 [(η x ) 2+δ ] < ∞ for some δ > 0. Let us state a result that is contained in Theorem 2.1 of [START_REF] Armendáriz | Zero-range condensation at criticality[END_REF].

Theorem 4.2 ([AGL13]

). Assume that b > 3 and L(•) ≡ c 1 in (4.5), and denote

ρ c = E ν 1 [η x ], σ 2 = Var ν 1 (η x ). Assume that N ≥ ρ c L and define γ ′ L via N = ρ c L + σ (b -3)L log L 1 + b 2(b -3) log log L log L + γ ′ L log L . (4.6) Then, if γ ′ := lim L→∞ γ ′ L ∈ [-∞, ∞], we have L M L S L -ρ c L S L = N w -→ (1 -p)δ 0 + pδ 1 , (4.7
)

with p = p γ ′ = (1 + σ b-1 (b-3) b/2 c 1 √ 2π e -(b-3)γ ′ ) -1 .
This result corresponds exactly to our Corollary 2.23 in the case where (2.28) holds with L(•) constant, see Example 2.14 (one can check that our definition of γn corresponds to the definition (4.6) of γ ′ L ). In fact, our Corollary 2.23 extends Theorem 4.2 to the case b = 3 (β = 2 in (2.28)) and allows a slowly varying function in (4.5). We refer to Example 2.17 for a definition of the threshold for the appearance of a condensate in that case; note that Rozovskii's condition (2.4) holds since η x is non-negative, see Example A.10 in the Appendix.

Example 4.3. Assume that w(n) ∼ c 1 n -3 in (4.5), or equivalently ν 1 (η x = n) ∼ cn -3 , which is a natural example with b = 3. One then has ν 1 (η x > n) ∼ c 2 n -2 , σ 2 (n) ∼ c log n, q(n) ∼ (2 log n) -1 and a n ∼ c 2 n log n. Assuming a slightly stronger condition on w(•), namely w(n) = cn -3 (1 + o( 1 log log n )) (which holds if g(k) = 1 + 3 k or g(k) = (k/(k -1))
3 ), we easily get that one can take exactly a n = c 2 n log n and Rozovskii's condition (2.4) holds, using Proposition A.3. Now we can use Example 2.17 to obtain that, setting x L = N -ρ c L and

γL := x 2 L 2a 2 L -log log a L - 3 2 log log log a L = x 2 L 2a 2 L -log log L + log 2 - 3 2 log log log L + o(1) , if in addition lim L→∞ γL = γ∞ ∈ [-∞, ∞]
, then (4.7) holds with p = (1 + 1 √ π e -γ∞ ) -1 . We can reframe this in the same way as in (4.6): setting γ ′ L to be such that

N = ρ c L + cL log L log log L 1 + 3 2 log log log L log log L + γ ′ L log log L , then if lim L→∞ γ ′ L = γ ′ , we obtain that (4.7) holds with p = (1 + 1 2 √ π e -γ ′ ) -1 .
Additionally, our Corollary 3.5 answers the question in the case where b ∈ [2, 3) (with

E ν 1 [η x ] < ∞ in the case b = 2).
It shows that a condensate that contains all the excess mass appears as soon as N -ρ c L ≫ a L , with a L the normalising sequence analogous to (1.5); this question was raised in [AGL13, p. 3477] (actually, the authors noticed that it mostly relied on Theorem 3.4 which was missing at the time).

To conclude, let us mention that the case b ∈ [1, 2] with E ν 1 [η x ] = ∞ is also considered in the recent paper [START_REF] Xu | Condensation of the invariant measures of the supercritical zero range processes[END_REF]: in that case, one needs to define a finite volume version of the critical density, ρ c,L , and the condensation phenomenon is shown to occur in some regime N ≫ ρ c,L L. We refer to [START_REF] Xu | Condensation of the invariant measures of the supercritical zero range processes[END_REF] for details.

Going further: scaling of the (recentered) maximum

Similarly to what is done for the overshoot in Corollary 2.12, we could try to obtain the correct scale of the maximum. We have in mind the following results, analogous to those in [AGL13, Thm. 2.1] (in the setting of Theorem 4.2), which identify the scaling limit of the condensate in the zero-range process (we present them here with a integral and a local conditioning and to a wider range of distributions). We state them as conjectures, since they are not simple consequences of results proved in this paper and would require some additional technical work that would lengthen the paper.

Conjecture 4.4. Let α = 2, let (a n ) n≥1 be a normalising sequence as in (1.5), and assume that F is intermediate regularly varying (see (2.11)). Let (x n ) n≥1 be a sequence such that lim n→∞ xn an = ∞ and let r n be defined as in (2.12).

• If lim n→∞ nF (xn) P(Sn-bn≥xn) < 1, we have

lim n→∞ P(M n ≤ t n | S n -b n ≥ x n , M n ≤ r n ) = e -t , (4.8)
for any sequence

(t n ) n≥0 such that lim n→∞ nF (t n ) = t ∈ [0, ∞]. • If lim n→∞ nF (xn)
P(Sn-bn≥xn) > 0, we have P(Sn-⌊bn⌋=xn) . We stress that the conjecture is actually only about the statement in (4.8), since (4.9) is a direct consequence of Theorem 2.10; we have kept the statement inside the conjecture since it gives a complete picture of the phenomenon.

L S n -b n -M n a n S n -b n ≥ x n , M n > r n w ---→ n→∞ N (0, 1). ( 4
The above statements can roughly be understood as follows:

• Conditioning on having a large deviation but no big-jump (M n ≤ r n ), the maximum behaves exactly as the maximum of i.i.d. random variables with law P i.e. without conditioning. Note that (4.8) is a unified way of considering convergence to extreme value distributions: if F (x) ∼ e -x (resp. F (x) ∼ x -β ), then t n = log n -log t + o(1) (resp. t n ∼ t -1/β n 1/β ) so one recovers the Gumbel (resp. Fréchet) distribution by a simple change of variable u = -log t (resp. u = t -1/β ).

• Conditioning on having a big jump (M n > r n ), the fluctuations of the maximum around its typical value S n -b n (≥ x n ) is Gaussian, on a scale a n .

In particular, Conjecture 4.4 sheds light on the law of the maximum (the condensate) in a regime where lim n→∞ nF (xn) P(Sn-bn≥xn) = s ∈ (0, 1): the conditional law of the maximum will be a non-trivial mixture of a scaled exponential and a (differently) scaled and translated Gaussian.

Multiple big-jumps

An interesting direction of research would be to generalise the results obtained in the present paper to a setting where the large deviation is obtained not by a single big jump but by multiple ones. This occurs for instance if the random variables are subject to a cutoff (or a dampening) which may depend on the number of variables in the sum, as considered in [START_REF] Kerriou | The fewest-big-jumps principle and an application to random graphs[END_REF]. The idea is to consider large deviation probabilities of the type

P(S n -b n ≥ x n , M n ≤ c n ) ,
where c n is a given cutoff. Alternatively, one may consider a triangular array of variables (ξ

(n) i ) 1≤i≤n
, with a heavy-tailed law which is truncated (or dampened) at a threshold c n . Then, if ξ has a heavy tail and x n grows sufficiently fast, one expects a "fewest-big-jumps" principle to replace the "one-big-jump" principle: the large deviation should be realised mostly thanks to k n = ⌈x n /c n ⌉ random variables being close to the cutoff c n . This is what is investigated in [START_REF] Kerriou | The fewest-big-jumps principle and an application to random graphs[END_REF]; let us also mention [CD19, Thm. 1.1] where a general upper bound for the local large deviation is given, with this underlying philosophy.

Several questions have been answered in [START_REF] Kerriou | The fewest-big-jumps principle and an application to random graphs[END_REF], making the fewest-big-jumps principle precise, but many problems remain open. For instance, can we obtain a result analogous to Propositions 2.8 and 2.16 if one adds a dampening to the law of ξ? More precisely, if one assumes that P(ξ = x) ∼ βx -(1+β) for some β ≥ 2, one could conjecture that, for some reasonable choices of the cutoff (c n ) n≥1 (e.g. c n = γn for some γ ∈ (0, 1)), one has roughly

P(S n -b n ≥ x n , M n ≤ c n ) ≈ P(S n -b n ≥ x n , M n ≤ r n ) + n k n P(ξ = c n ) kn ≈ Φ x n a n + n k n P(ξ = c n ) kn ,
where k n = ⌈x n /c n ⌉. A natural question would then be to understand the transition between a collective (Gaussian) and a few-individuals (multiple big-jumps) behaviour, analogously to Examples 1.1 and 2.14. The same question can be considered in the local large deviation setting.

Let us conclude by mentioning that in the context of the zero-range processes, one could consider a model with a saturation threshold c L , meaning that a site cannot host (much) more than c L particles. This corresponds to taking the function g in Section 4.2 as being L-dependent and dampening the effective weight w(n) at the threshold c L ; for instance taking g(n) = ∞ for n > c L , one gets that w(n) = 0 for all n ≥ c L . Proving the "fewest big-jumps" principle, i.e. obtaining sharp asymptotics of the type

P(S L = N, M L ≤ c L ) ∼ L k L P(ξ = c L ) k L with k L = ⌈L/c L ⌉,
could then possibly be translated into a statement that the zero-range process possesses k L condensates of size close to c L .

Proofs of the large deviations and conditional laws results

Before we start the proof of our results, let us introduce some notation. We use the same letter c, c 0 , c 1 , c 2 etc, to denote constants at various places, but they may refer to different values. For positive functions f, g we write f (y) ≍ g(y) if there exist c > 0, C > 0 such that cg(y) ≤ f (y) ≤ Cg(y) for y ≥ 1.

Also, to simplify the statements, we will assume that µ = E[ξ] = 0. Recall the definition (1.3) of σ 2 (x) = E[ξ 2 1 |ξ|≤x ]; as we assume in this section that ξ is in the domain of attraction of the normal law, σ 2 (x) is slowly varying at ∞.

Some preliminary estimates

Let us now collect some estimates that will be useful in the rest of the paper: bounds on P(|ξ| > x) and truncated moments that involve σ 2 (x); a Fuk-Nagaev type inequality for P(S n ≥ x, M n ≤ y) in the case α = 2.

Estimating the tail with the truncated second moment

For x > 0, let us set

q(x) := x 2 σ 2 (x) P(|ξ| > x) ,
and q * (x) := sup y≥x q(y) and q(x) := 1 x x 0 q(t)dt.

Claim 5.1. If x → σ 2 (x) is slowly varying, then we have lim x→∞ q(x) = 0; a direct consequence is that lim x→∞ q * (x) = lim x→∞ q(x) = 0. Also, there is a constant c > 0 such that

E[|ξ|1 {|ξ|>x} ] ≤ c q * (x)σ 2 (x)x -1 ,
(5.1)

E[|ξ| 3 1 {|ξ|≤x} ] ≤ c q(x)σ 2 (x)x .
(5.2)

Proof. For the first part of the statement, the fact that lim x→∞ q(x) = 0 is recalled in (1.4), see e.g. [Fel08, Ch. IX.8, Eq. (8.5)].

To prove (5.1), we write

E[|ξ|1 {|ξ|>x} ] = xP(|ξ| > x) + ∞ x P(|ξ| > t)dt ≤ q(x)σ 2 (x) x + q * (x) ∞ x σ 2 (t)t -2 dt ,
where we have used that P(|ξ| > t) = q(t)σ 2 (t)t -2 and the definition of q * . Then, using the properties of regularly varying functions, the integral is asymptotically equivalent to σ 2 (x)x -1 , which gives the desired bound.

To prove (5.2), write

E[|ξ| 3 1 {|ξ|≤x} ] = 3 ∞ 0 t 2 P(|ξ|1 {|ξ|≤x} > t)dt ≤ 3 x 0 t 2 P(|ξ| > t)dt ≤ 3σ 2 (x) x 0 q(t)dt ,
where we used the fact that t → σ 2 (t) is non-decreasing. With the definition of q(x), this gives the desired conclusion.

A Fuk-Nagaev inequality

Lemma 5.2. Assume that µ = 0 and that x → σ 2 (x) is slowly varying. Then there is some r 0 > 0 such that for any x, y ≥ r 0 P(S n ≥ x, M n ≤ y) ≤ e x/y 1 + xy nσ 2 (y) -x/y . Proof. We use [Nag79, Theorem 1.2] with t = 2, to get the following: ≤ 0, which concludes the proof.

P(S n ≥ x, M n ≤ y) ≤ e x y 1 + xy nσ 2 (y) -x y + nµ(y) y -nσ 2 (y)

Proof of Proposition 2.4

We present here a proof in a slightly more general setup: we will replace the constraint M n ≤ r n by M n ≤ ar n , for some fixed a > 0. We stated Proposition 2.4 only for a = 1 but we also need the result in the proof of Proposition 2.8 with a = 4, see (5.19) below. Let us start with estimates in the case of a generic r, which will later on be replaced by r = ar n .

Estimates on the tilted measure

Let us redefine P u from (2.15) with a general r instead of r = r n (we keep the same notation for simplicity). For any u ≥ 0 and r > 0 dP u dP (x) = dP (5.3) Claim 5.3. Let 0 < c < C be fixed constants. Assume µ = 0. If x → σ 2 (x) is slowly varying at infinity, then as r → ∞, uniformly for c r ≤ u ≤ C r , we have

M (u) -1 = o(r -1 σ 2 (r)) ; (i) M ′ (u) = (1 + o(1))u σ 2 (r) ; (ii) M ′′ (u) = (1 + o(1)) σ 2 (r) . (iii)
We also have that M ′′ (u) ≥ (1 + o(1))σ 2 (r) for any 0 ≤ u ≤ C r . Finally, setting q * (x) := q * (x) + q(x), we have

E |ξ| 3 e uξ 1 (-∞,r] (ξ) ≤ c 0 q * (r)rσ 2 (r) . (iv) Proof. Define, for k ≥ 0, m k (r) = E[ξ k 1 [-r,r] (ξ)]; in particular we have σ 2 (r) = m 2 (r).
Item (i). We have

M (u) = E[e uξ 1 {|ξ|≤r} ] + E[e uξ 1 {ξ<-r} ] .
(5.4)

We can estimate the first term as follows: using that |uξ| ≤ C for any |ξ| ≤ r, expanding the exponential, we get

E[e uξ 1 {|ξ|≤r} ] = m 0 (r) + u m 1 (r) + O(u 2 m 2 (r)) .
Using that u ≤ C/r, we get that the last term is bounded by a constant times r -2 σ 2 (r) = o(r -1 σ 2 (r)). Now, thanks to Claim 5.1, we obtain the following estimates (using that µ = 0 to get that m 1 (r) = -E[ξ1 {|ξ|>r} ]):

m 0 (r) = 1 -P(|ξ| > r) = 1 -o(r -2 σ 2 (r)) , |m 1 (r)| ≤ E[|ξ|1 {|ξ|>r} ] = o(r -1 σ 2 (r)) .
Therefore, we obtain E[e uξ 1 {|ξ|≤r} ] = o(r -1 σ 2 (r)).

For the second term in (5.4), recalling Claim 5.1, we have

0 ≤ E[e uξ 1 {ξ<-r} ] ≤ ∞ k=0 e -c2 k P(ξ ∈ [-2 k+1 r, -2 k r)) ≤ ∞ k=0 e -c2 k P(|ξ| > 2 k r)) = ∞ k=0 e -c2 k q(2 k r)σ 2 (2 k r) 2 2k r 2 ) ≤ r -2 q * (r) ∞ k=0 2 2k e -c2 k σ 2 (2 k r) ≤ Cr -2 q * (r)σ 2 (r) ,
where we used Potter's bound (see [BGT89, §1.5.4]) for the last inequality, to get that σ 2 (2 k r) ≤ 2 k σ 2 (r). All together, we obtain (i).

Item (ii).

Let us now turn to M ′ (u). Again, we write M ′ (u) as

M ′ (u) = E[e uξ ξ1 {|ξ|≤r} ] + E[e uξ ξ1 {ξ<-r} ] .
For the first term, expanding the exponential, we get E[e uξ ξ1 {|ξ|≤r} ] = m 1 (r) + um 2 (r) + O(u 2 m 3 (r)) .

As above, we get that m 1 (r) = o(r -1 σ 2 (r)) and thanks to Claim 5.1-(5.2) we also have u 2 m 3 (r) = O(rq(r)σ 2 (r)) = o(r -1 σ 2 (r)). For the second term, recalling Claim 5.1, we get similarly as above that -cr -1 q * (r)σ 2 (r) ≤ E[e uξ ξ1 {ξ<-r} ] ≤ 0 , which concludes the proof for (ii). Item (iii). As above, we write

M ′′ (u) = E[e uξ ξ 2 1 {|ξ|≤r} ] + E[e uξ ξ 2 1 {ξ<-r} ] .
Expanding the exponential, we get that the first term is σ 2 (r)+O(r -1 m 3 (r)), with |m 3 (r)| ≤ rq(r)σ 2 (r) thanks to Claim 5.1. Since the second term is non-negative, this gives the general lower bound M ′′ (u) ≥ (1 + o(1))σ 2 (r). When u ≥ c/r, the second term is treated as above: using Claim 5.1 and Potter's bound, we get

0 ≤ E[e uξ ξ 2 1 {ξ<-r} ] ≤ ∞ k=1 2 2k+2 r 2 e -c2 k P(ξ ∈ [-2 k+1 r, -2 k r)) ≤ Cq * (r) ∞ k=1 2 5k e -c2 k σ 2 (r) .
The last term is bounded by a constant times q * (r)σ 2 (r) = o(σ 2 (r)). Then we can conclude the proof for (iii).

Item (iv).

We have

E |ξ| 3 e uξ 1 (-∞,r] (ξ) ≤ E[e uξ |ξ| 3 1 {ξ≤r} ] + E[e uξ |ξ| 3 1 {ξ≤-r} ] .
The first term is bounded by a constant times E[|ξ| 3 1 {|ξ|≤r} ] ≤ crq(r)σ 2 (r) thanks to Claim 5.1. The second term is treated as above and is bounded by a constant times rq * (r)σ 2 (r).

We now present a corollary, in the case where r = ar n for some fixed constant a > 0. Let us set a few notations first. From now on, we set, P u := P (arn) u and M (u) := M arn (u) for any u ≥ 0, with the notation from (5.3). Then, recall the definition of λ(t)

:= m -1 (t), where m(u) = E u [ξ] = M ′ (u) M (u) (recall that u → m(u) is increasing). Hence, λ(t) is such that E λ(t) [ξ] = t.
Note that for λ(t) to be well-defined and non-negative, we must take t ≥ m(0

) := E[ξ | ξ ≤ ar n ]. Corollary 5.4. Assume µ = 0. If lim n→∞ xn n = 0, then m(0) = o( xn n ). Let 0 < c ≤ d < ∞ be fixed, then uniformly for s ∈ [c, d] we have λ(s xn n ) = (1 + o(1)) s rn . As a consequence, we have λ(t) ∈ [(1 + o(1)) c rn , (1 + o(1)) d rn ] uniformly for t ∈ [c xn n , d xn n ]. Proof. Note that since E[ξ] = 0, we have m(0) = E[ξ | ξ ≤ ar n ] = - 1 P(ξ ≤ ar n ) E[ξ1 ξ>arn ] ≤ 0 . ( 5

Estimate of the relative entropy: proof of Lemma 2.6

Before we prove Proposition 2.4, as a warm up computation, we analyse the entropy H(•) defined in (2.18), that is we prove Lemma 2.6 above 4 . We present the proof with ar n instead of r n in the definition (2.18) of H, that is we take H(t) = H arn (t).

Proof of Lemma 2.6. First of all, notice that

H ′ (t) = -λ ′ (t) M ′ (λ(t)) M (λ(t)) + λ(t) + tλ ′ (t) = λ(t) , since M ′ (λ(t)) M (λ(t)) = m(λ(t)) = t
, by definition of λ(t), see (2.17). Recalling that m 0 = m(0), we get that H(m 0 ) = -log M (0) = -log P(ξ ≤ ar n ) so

H(m 0 ) ∼ F (ar n ) ∼ q(ar n ) σ 2 (r n ) a 2 r 2 n = o x 2 n n 2 σ 2 (r n ) .
(5.6) 4 The corresponding result is proved in p. 107 of [START_REF] Doney | A large deviation local limit theorem[END_REF], where it is claimed that nH( xn n ) =

x 2 n 2a 2 n + o(1); we actually realised that an argument is missing in [START_REF] Doney | A large deviation local limit theorem[END_REF] (one needs to control the error term in the variance of the tilted law). In our Lemma 2.6, we have o(1) as a factor: to obtain nH( xn n ) = Here we have used the fact that lim x→∞ q(x) = 0 and the definition (2.13) of r n . Moreover, we have H ′ (m 0 ) = λ(m 0 ) = 0. By Taylor's theorem, we have

H x n n -H(m 0 ) = xn/n m 0 x n n -t H ′′ (t) dt.
Using that H ′ (t) = λ(t), we get that

H ′′ (t) = 1 m ′ (λ(t)) = M (λ(t)) 2 M ′′ (λ(t))M (λ(t)) -M ′ (λ(t)) 2 = 1 E λ(t) [ξ 2 ] -t 2 . Now, we have that 0 ≤ λ(t) ≤ (1 + o(1)) 1
rn for all t ∈ [m 0 , xn n ], thanks to Corollary 5.4, which gives

E λ(t) [ξ 2 ] = M ′′ (λ(t)) M (λ(t)) ≥ (1 + o(1))σ 2 (r n ) ,
thanks to Claim 5.3 (see the lower bound below (iii)). Note that σ 2 (r n )/( xn n ) 2 → ∞, due to (2.13). Therefore, using (5.6) and that m 0 = o( xn n ) from Corollary 5.4, we have

H x n n ≤ (1 + o(1)) 1 2σ 2 (r n ) x n n -m 0 2 + H(m 0 ) = (1 + o(1)) x 2 n 2n 2 σ 2 (r n ) .
To get a lower bound, we use the fact that for t ∈ [ε xn n , xn n ] we have c rn ≤ λ(t) ≤ C rn for some positive constants c, C, see Corollary 5.4. Hence, thanks to Claim 5.3, we get that

H ′′ (t) = (1 + o(1))σ 2 (r n ) -1 uniformly for t ∈ [ε xn n , xn n ], using also that t 2 = o(σ 2 (r n ))
. We end up with

H x n n ≥ (1 + o(1)) 1 σ 2 (r) x/n εxn/n x n n -t dt + H(m 0 ) ≥ (1 -2ε) x 2 n 2n 2 σ 2 (r n ) .
This concludes the proof.

Completion of the proof of Proposition 2.4

Recall that we assume µ = 0 in this section. We fix λ = λ n := λ( xn n ) so that E λ [S n ] = nm(λ) = x n . We can write

P(S n ≥ x n , M n ≤ ar n ) = M (λ) n E λ e -λSn 1 {Sn≥xn} = e n[log M (λ)-λm(λ)] E λ e -λ(Sn-xn) 1 {Sn-xn≥0} = e -nH( xn n ) E λ e -λ(Sn-xn) 1 {Sn-xn≥0} .
(5.7) In this step of the proof of Proposition 2.4, we simply need to control the expectation on the r.h.s in the third equality. The behaviour of nH( xn n ) has been studied in Lemma 2.6.

We can write the last term as E λ [e -λ Sn 1 Sn≥0 ], where Sn = n i=1 ξi with ξi = ξ i -xn n . In particular, using Claim 5.3, we have

E λ [ ξi ] = 0 , E λ [ ξ2 i ] ∼ σ 2 (r n ) , E λ [| ξi | 3 ] ≤ c a q * (r n )r n σ 2 (r n ) .
(5.8) Indeed, using Corollary 5.4 we have λ = λ( xn n ) ∼ 1 rn , which yields M (λ) → 1, see (i) in Claim 5.3. Then applying (ii) and (iii) in Claim 5.3, we have Var λ

[ ξi ] = Var λ [ξ i ] = M ′′ (λ) M (λ) 2 ∼ σ 2 (r n ) and E λ [| ξi | 3 ] ≤ 4E λ [|ξ i | 3 ] + 4 xn n 3 ≤ cq * (r n )r n σ 2 (r n ). The fact that E λ [ξ] = xn
n is by the definition of λ. This gives (5.8).

Thanks to the Berry-Esseen bound, this gives that uniformly for y ∈ R,

P λ 1 nσ 2 (r n ) S n ≤ y -Φ(y) ≤ C nσ 2 (r n ) E λ [| ξi | 3 ] σ 2 (r n ) ≤ cq * (r n ) r n nσ 2 (r n ) , (5.9)
which goes to 0. Indeed, note that since r n ∼ n xn σ 2 (r n ) and √ n ∼ a n / σ 2 (a n ), we have

r n nσ 2 (r n ) ∼ a n x n σ 2 (r n ) σ 2 (a n ) .
(5.10)

In particular, we get that

rn √ nσ 2 (rn)
≤ c an xn : this is bounded by a constant (since we are considering x n ≥ a n ) and goes to 0 if lim n→∞ xn an = ∞. Since q * (r n ) → 0, (5.9) therefore shows that 1 √ nσ 2 (rn) Sn converges under P λ to a standard Gaussian. Additionally, we can use the Berry-Esseen bound (5.9) to get that

E λ [e -λ Sn 1 Sn≥0 ] -E[e -λ √ nσ 2 (rn)Z 1 {Z≥0} ] = O q * (r n ) r n nσ 2 (r n ) = o(1) .
(5.11) Indeed, we can write

E λ [e -λ Sn 1 Sn≥0 ] = 1 0 P λ (t ≤ e -λ Sn ≤ 1)dt = 1 0 P λ 0 ≤ 1 nσ 2 (r n ) S n ≤ log(1/t) λ nσ 2 (r n ) dt = 1 0 P λ 0 ≤ Z ≤ log(1/t) λ nσ 2 (r n ) dt + O q * (r n ) r n nσ 2 (r n ) ,
where we have used (5.9) for the last equality. Now, with the same manipulation, the last integral is equal to

E[e -λ √ nσ 2 (rn)Z 1 {Z≥0} ]. Now, if lim n→∞ xn an = ∞, then rn √ nσ 2 (rn)
goes to 0, recall (5.10), so λ nσ 2 (r n ) ∼ √ nσ 2 (rn) rn goes to ∞: using that E[e -uZ 1 {Z≥0} ] = e u 2 /2 Φ(u), we obtain

E[e -λ √ nσ 2 (rn)Z 1 {Z≥0} ] ∼ 1 √ 2π r n nσ 2 (r n ) ∼ 1 √ 2π a n x n σ 2 (r n ) σ 2 (a n ) ,
so that thanks to (5.11) we end up with

E λ [e -λ Sn 1 Sn≥0 ] ∼ 1 √ 2π a n x n σ 2 (r n ) σ 2 (a n ) .
(5.12)

If on the other hand x n = O(a n ), then recalling (5.10) and the fact that

σ 2 (r n ) ∼ σ 2 (a n ) if a n ≤ x n ≤ Ca n , we get that λ n := λ nσ 2 (r n ) = (1 + o(1)) xn rn = (1 + o(1)) xn an
, where in the last equality we used

x n r n ∼ nσ 2 (r n ) ∼ nσ 2 (a n ) ∼ a 2
n . Furthermore, we have

E[e -λnZ 1 {Z≥0} ] = e λ 2
n /2 Φ(λ n ). We therefore end up with

P(S n ≥ x n , M n ≤ ar n ) = (1 + o(1))e -nH( xn n ) e λ 2 n /2 Φ(λ n ) = (1 + o(1))Φ x n a n ,
where for the last identity we used Lemma 2.6 to get nH

( xn n ) = x 2 n 2a 2 n + o(1) = 1 2 λ 2 n + o(1) (since x n = O(a n )).

Proof of Proposition 2.8

5.3.1 Preliminary: a tail estimate using intermediate regular variation of F Lemma 5.5. Let α = 2, let (a n ) n≥1 be a normalising sequence as in (1.5) and assume that µ = 0 and that F is intermediate regularly varying (recall (2.11)). Let (C n ) n≥1 be such that lim n→∞ nF (C n ) = 0. Then, we have lim n→∞ C n = ∞ and

lim ε→0 lim n→∞ sup x≥Cn P(S n ≥ x, M n ≥ (1 -ε)x) P(S n ≥ x, M n ≥ x) -1 = 0.
(5.13)

In particular, if lim n→∞ xn an = ∞ holds, then we can use (x n ) n≥1 in place of (C n ) n≥1 and

P(S n ≥ x n , M n ≥ x n ) ∼ P(M n ≥ x n ) ∼ nF (x n ) .
(5.14)

Proof. The intermediate regular variation of F and the definition of (C n ) n≥1 imply that lim n→∞ C n = ∞ and F (C n ) > 0 for all n. Next, we note that

P(M n ≥ x) ∼ nF (x) , as n → ∞ (5.15)
uniformly for x ≥ C n . Note also that given nF (C n ) = o(1) and x ≥ C n , we have

P(S n ≥ x, M n ≥ (1 -ε)x) -P(S n ≥ x, M n ≥ x) ≤ P((1 -ε)x ≤ M n < x) , with P((1 -ε)x ≤ M n < x) = P(M n ≥ (1 -ε)x) -P(M n ≥ x) = (1 + o(1))n F ((1 -ε)x) -F (x) + o(1)nF ((1 -ε)x)
where the second equality is due to (5.15). Moreover,

P(S n ≥ x, M n ≥ x) ≥ nP(ξ ≥ x)P(S n-1 ≥ 0) - n 2 P(ξ 1 ≥ x, ξ 2 ≥ x) ≥ (1 + o(1))nF (x)P(S n-1 ≥ 0) -(1 + o(1))(nF (x)) 2 ≥ n 4 F (x) ,
where the last two inequalities hold for n large enough, uniformly for x ≥ C n . Indeed, we have that lim n→∞ P(S n-1 ≥ 0) = 1 2 by the Central Limit Theorem and (nF (x)) 2 = o(nF (x)) for x ≥ C n . Combining the above three displays,

P(S n ≥ x, M n ≥ (1 -ε)x) P(S n ≥ x, M n ≥ x) -1 ≤ 8 sup x≥Cn F ((1 -ε)x) F (x) -1 + o(1) sup x≥Cn F ((1 -ε)x) F (x) , (
F ((1 -ε)x) F (x) ≤ (1 + sup x≥Cn δ (1-ε)x )κ 1 1 -ε .
Note that sup x≥Cn δ (1-ε)x converges to 0 as n → ∞, and κ( 1 1-ε ) converges to 1 as ε → 0. Therefore, (5.16) entails (5.13).

For the second part of the lemma, the fact that (x n ) n≥1 is a candidate for (C n ) n≥1 is due to (1.4) and (1.5). Then, applying (5.15), we get

P(S n ≥ x n , M n ≥ x n ) ≤ P(M n ≥ x n ) ∼ nF (x n ) .
Hence it only remains to prove a matching lower bound. For any ε > 0, we have that

P(S n ≥ x n , M n ≥ x n ) ≥ P(S n ≥ x n , M n ≥ (1 + ε)x n ) ≥ nP ξ > (1 + ε)x n P S n-1 ≥ -εx n - n 2 P ξ 1 ≥ (1 + ε)x n , ξ 2 ≥ (1 + ε)x n .
Since lim n→∞ P(S n-1 ≥ -εx n ) = 1 for any ε > 0, because lim n→∞ xn an = ∞, we get that the lower bound is asymptotically equivalent to nF ((1+ε)x n ). Then, thanks to the intermediate regular variation of F , we get lim ε↓0 lim inf x→∞

F ((1+ε)x) F (x)
= 1, which concludes the proof.

Completion of the proof of Proposition 2.8

Recall that we assume that F is intermediate regularly varying and also that µ = 0. We will focus on the case where lim n→∞ xn an = ∞ and lim n→∞ xn n = 0; recall also that (2.13) holds. (We briefly discuss at the end of the proof for the case where lim inf n→∞ xn n > 0, which can also be found e.g. in [START_REF] Denisov | Large deviations for random walks under subexponentiality: the big-jump domain[END_REF].) Using Lemma 5.5, we simply need to show that

P(S n ≥ x n , M n > r n ) ∼ P(S n ≥ x n , M n > (1 -ε)x n ),
(5.17)

for any ε ∈ (0, 1). We fix ε ∈ (0, 1) and write the probability as

Q 1 + Q 2 + Q 3 + Q 4 , with Q 1 = P(S n ≥ x n , M n > (1 -ε)x n ) , Q 2 = P(S n ≥ x n , εx n < M n ≤ (1 -ε)x n ) , Q 3 = P(S n ≥ x n , 4r n < M n ≤ εx n ) , Q 4 = P(S n ≥ x n , r n < M n ≤ 4r n ) .
By Lemma 5.5, Q 1 ∼ nF (x n ). We show next that all other terms are negligible compared to nF (x n ). The term Q 2 can be estimated as follows: we have that

Q 2 ≤ nP εx n < ξ ≤ (1 -ε)x n P S n-1 ≥ εx n . Using Potter's bound (A.2) (see [Cli94, Thm. 2.3]), we get that P(εx n < ξ ≤ (1 -ε)x n ) ≤ F (εx n ) ≤ c ′ κ(1/ε)F (x n )
. Also, since lim n→∞ xn an = ∞ holds, by the convergence in distribution of (1.1) (recall b n = 0 since µ = 0) we get that lim n→∞ P(S n-1 ≥ εx n ) = 0 for any fixed ε > 0. All together, this shows that lim n→∞

Q 2 nF (xn) = 0. Note that we have r n a n σ 2 (a n ) σ 2 (r n ) ∼ a n x n , and lim n→∞ r n a n = 0 if lim n→∞ x n a n = ∞. (5.18)
The first result is due to (2.13) and the definition of a n , see (1.5). The second result relies on the first one and uses that σ 2 (•) is slowly varying. Then, since lim n→∞ xn an = ∞ we have lim n→∞ xn rn = ∞. We can now bound Q 3 as follows (omitting the floor function ⌊•⌋ to lighten notation):

Q 3 = log 2 ( xn rn )-1 k=log 2 ( 1 ε ) P S n ≥ x n , M n ∈ (2 -k-1 x n , 2 -k x n ] ≤ log 2 ( xn rn )-1 k=log 2 ( 1 ε ) nF (2 -k-1 x n )P S n-1 ≥ 1 2 x n , M n-1 ≤ 2 -k x n .
Then, using Potter's bound (A.2), we have that nF (2 -k-1 x n ) ≤ c 0 κ(2 k+1 )nF (x n ) uniformly in k ≥ 1 for some c 0 > 0, for n large enough. Using the Fuk-Nagaev inequality of Lemma 5.2, we get that

Q 3 nF (x n ) ≤ c 0 log 2 ( xn rn )-1 k=log 2 ( 1 ε ) κ(2 k+1 )e 2 k 1 + 2 -k x 2 n nσ 2 (2 -k x n ) -2 k ≤ c 0 log 2 ( xn rn )-1 k=log 2 ( 1 ε ) κ(2 k+1 )e 2 k 11 4 -2 k
, where we have used that xny nσ 2 (y) ≥ 7/4 uniformly for y ≥ 2r n (provided that n is large enough). Indeed, we have that xny nσ 2 (y) ≥ (1 + o(1))2 xnrn nσ 2 (rn) uniformly for y ≥ r n , and the lower bound goes to 2 as n → ∞. We therefore have shown that

Q 3 nF (x n ) ≤ c 0 ∞ k=log 2 ( 1 ε ) κ(2 k+1 )(11/4e) -2 k -→ 0, as ε ↓ 0.
The convergence is due to the definition of κ(•) and the fact that 11 > 4e. It remains to treat Q 4 . We apply again Potter's bound (A.2) to obtain that

Q 4 ≤ nF (r n )P S n-1 ≥ 1 2 x n , M n-1 ≤ 4r n ≤ cnF (x n )κ x n r n P S n-1 ≥ 1 2 x n , M n-1 ≤ 4r n .
For the last probability, the Fuk-Nagaev inequality of Lemma 5.2 is not enough, because of the first factor e xn/rn . However, thanks to Proposition 2.4 (one can replace n by n -1, x n by 1 2 x n and r n by 4r n , that changes only some constants, see in particular (5.7)), we have

P S n-1 ≥ x n /2, M n-1 ≤ 4r n ≤ e -nH( xn 2n ) ≤ e -c
x 2 n nσ 2 (rn) ≤ e -c xn rn , (5.19) see (5.7) and Lemma 2.6; we also have used that nσ 2 (r n ) ∼ r n x n . Hence, we get that

Q 4 nF (x n ) ≤ cκ x n
r n e -cxn/rn , which goes to 0 since x n /r n → ∞ as noticed above. All together, this concludes the proof of Proposition 2.8 in the case where lim n→∞ xn n = 0 and lim n→∞ xn an = ∞. In the case where lim inf n→∞ xn n > 0, we get that r n = O(1). Then, by exactly the same proof we can control the terms Q 1 , Q 2 , Q 3 . It then simply remains to show that for any fixed R > 0, Q4 :

= P(S n ≥ x n , M n ≤ R) is negligible compared to Q 1 ∼ nF (x n ).
But this is simply a large deviation estimate for bounded random variables (with negative mean): we get that Q4 ≤ e -cxn (note that Q4 = 0 if x n ≥ Rn), which is indeed negligible compared to nF (x n ) since F decays slower than any exponential function.

Proofs of Theorems 2.10, of Corollary 2.12 and of Corollary 2.22

Proof of Theorem 2.10. By Proposition 2.8 and equation (5.14), we have that

P(S n -b n ≥ x n , M n > r n ) ∼ P(S n -b n ≥ x n , M n ≥ x n ) ∼ P(M n ≥ x n ).
Then we apply (4.2) to obtain (2.21). The fact lim n→∞ P(M n-1 ≤ r n ) = 0 entails (2.22). The convergence (2.23) is a direct consequence of (2.21) and (2.22). The last result is obtained simply by applying Theorem 2.1. This completes the proof.

Proof of Corollary 2.12. Assume µ = 0 for simplicity so that in particular b n ≡ 0. Recall we assume lim n→∞ xn an = ∞. Item 1. Recall in this case we assume lim n→∞ xn n = 0. For A n ∈ B(R n ) we have, using a change of measure as in (5.7),

P (ξ 1 , . . . , ξ n ) ∈ A n S n ≥ x n , M n ≤ ar n = P (ξ 1 , . . . , ξ n ) ∈ A n , S n ≥ x n , M n ≤ ar n P(S n ≥ x n , M n ≤ ar n ) = E λn e -λn(Sn-xn) 1 {Sn-xn≥0} 1 An (ξ 1 , . . . , ξ n ) E λn e -λn(Sn-xn) 1 {Sn-xn≥0}
.

Note that (5.9) together with the argument for (5.11) actually gives the quantitative bound sup

0≤b 1 <b 2 ≤∞ E λn [e -λn(Sn-xn) 1 b 1 ≤Sn-xn<b 2 ] -E[e -λn √ nσ 2 (rn)Z 1 b 1 ≤Z≤b 2 } ] = O q * (r n ) r n nσ 2 (r n ) . (5.20) Using E λn e -λn(Sn-xn) 1 {Sn-xn≥0} ∼ 1 √ 2π an xn σ 2 (rn)
σ 2 (an) (see (5.12)) and (5.10), together with the fact that q * (r n ) → 0, then yields

P (ξ 1 , . . . , ξ n ) ∈ A n S n ≥ x n , M n ≤ ar n = λ n 2πnσ 2 (r n ) E λn e -λn(Sn-xn) 1 {Sn-xn≥0} 1 An (ξ 1 , . . . , ξ n ) 1 + o(1) .
(5.21)

In fact, more quantitatively, the o(1) term above is O(q * (r n )).

Put

u n := λ n nσ 2 (r n ) ∼ 1 r n nσ 2 (r n ) ∼ x n /r n .
Notice that combining λ n = λ(x n /n) ∼ 1/r n from Corollary 5.4 with (5.10) shows that lim n→∞ u n diverges. For 0 ≤ z 1 < z 2 < ∞ let

A n = {(y 1 , . . . , y n ) ∈ R n : x n + u n z 1 ≤ y 1 + • • • + y n < x n + u n z 2 } , (5.22) so that {(ξ 1 , . . . , ξ n ) ∈ A n } = {z 1 ≤ (S n -x n )/u n < z 2 }.
Then from (5.21) and (5.20), using the definition of u n , we get

P z 1 ≤ (S n -x n )/u n < z 2 S n ≥ x n , M n ≤ ar n = E[ √ 2π u n e -unZ 1 z 1 un≤Z<z 2 un} ] 1 + o(1) = z 2 un z 1 un e -unz-z 2 /2 u n dz 1 + o(1) ---→ n→∞ e -z 1 -e -z 2 ,
which concludes the proof of item 1.

Item 2. Recall in this case we assume that F is intermediate regularly varying. The arguments for Proposition 2.8 show that P(S n ≥ x n , M n > r n ) ∼ nF (x n ), recall (5.17). Applying this again with x n replaced by x ′ n ≥ x n yields (2.25).

Proof of Corollary 2.22. By Proposition 2.8, we have that

lim n→∞ P(M n ≥ x n | S n -b n ≥ x n ) = s and lim n→∞ P(M n ≤ r n | S n -b n ≥ x n ) = 1 -s . Conditionally on {S n -b n ≥ x n , M n ≥ x n }, if we write S n = M n + S ′ n , then S ′ n -b n
is of order a n thanks to (2.23) and (1.1): since x n /a n → ∞, we obtain that

L M n S n -b n S n -b n ≥ x n , M n ≥ x n w -→ δ 1 .
On the other hand, since we have that r n /x n → 0 (using that x n /a n → ∞), we get

L M n S n -b n S n -b n ≥ x n , M n ≤ r n w -→ δ 0 .
This concludes the proof.

6 Proofs of the local versions of the theorems 6.1 Proof of Proposition 2.15

Recall x n ≥ a n with lim n→∞ xn n = 0. For simplicity, we also assume µ = 0. Analogous to (5.7), we have that

P(S n = x n , M n ≤ ar n ) = e -nH( xn n ) P λ S n = x n with λ = λ( xn n ).
Then, we can apply a local limit theorem to estimate P λ S n = x n ; in particular, since the distribution P λ depends on n and x n , we need explicit estimates on the rate of convergence. Several results exist in this direction, such as [Muk92, Thm. 5] or [GW17; SN21], but none of these appear to give a sufficient bound for our purpose. Instead, we start from [Don97, Lem. 3]: setting σ 2 λ := Var λ (ξ) and

ν λ := E λ [|ξ -xn n | 3 ], it gives that √ nσ λ P λ (S n = x n ) - 1 √ 2π ≤ C ν λ √ nσ 3 λ + 2 √ nσ λ π ℓ e -n(1-|ϕ λ (t)|) dt , (6.1)
with ℓ := σ 2 λ 4ν λ and ϕ λ (t) = E λ (e itξ ) the characteristic function of ξ under P λ . Since we have that σ 2 λ ∼ σ 2 (r n ), see the first two results in (5.8), we only have to show that both terms on the r.h.s. of (6.1) go to zero.

For the first term, we have from (5.8) that

ν λ √ nσ 3 λ ≤ C q * (r n )r n nσ 2 (r n ) ,
which goes to 0 as proved for the last term in (5.9).

For the remaining term, notice that thanks to (5.8) we have that ℓ = σ 2 λ 4ν λ ≥ ℓ n := c q * (rn)rn and σ 2 λ ∼ σ 2 (r n ). We fix ε > 0 (small) and we need to show that both of the following terms go to 0:

I 1 = nσ 2 (r n ) ε ℓn e -n(1-|ϕ λ (t)|) dt , I 2 = nσ 2 (r n ) π ε e -n(1-|ϕ λ (t)|) dt .
Let us now estimate 1 -|ϕ λ (t)|. Since ξ has maximum span 1 (equivalent to (S n ) being aperiodic; see the definition of maximum span in [START_REF] Gnedenko | Limit distributions for sums of independent random variables[END_REF]), we can choose some K > 0 such that the law of ξ restricted to {-K, . . . , K} has maximum span 1. Then, it is standard to get that for any 0 < ε < 1, there is a constant c 0 > 0 (that depends on K, ε) such that for any t ∈ [ε, π] E e itξ |ξ| ≤ K ≤ 1 -c 0 , see e.g. [GK54, §14, Cor. 2 to Thm. 5]. Now, turning back to ϕ λ , we have that, for n large enough so that K ≤ r n ,

|ϕ λ (t)| = E λ e itξ 1 {|ξ|≤K} + E λ e itξ 1 {|ξ|>K} ≤ E λ e itξ 1 {|ξ|≤K} + P λ (|ξ| > K) .
Now, recalling that λ → 0, we can apply Taylor expansion to e λξ on the event {|ξ| ≤ K}: for n large enough, we obtain

M (λ) E λ e itξ 1 {|ξ|≤K} = E e itξ+λξ 1 {|ξ|≤K} ≤ P(|ξ| ≤ K) E λ e itξ |ξ| ≤ K + 2λK ≤ (1 -c 0 2 )P(|ξ| ≤ K) .
Now, since λ → 0, we also easily get that M (λ)P(|ξ| ≤ K) = (1 + o(1))P λ (|ξ| ≤ K), so we obtain that for n large enough

|ϕ λ (t)| ≤ (1 -c 0 3 )P λ (|ξ| ≤ K) + P λ (|ξ| > K) = 1 -c 0 3 P λ (|ξ| ≤ K) ≤ 1 -c ′
, where we also have used for the last inequality that

P λ (|ξ| ≤ K) = (1 + o(1))P(|ξ| ≤ K), since λ → 0, M (λ) → 1. All together, we get that there is a constant c ′ = c ′ (ε) > 0 such that 1 -|ϕ λ (t)| ≥ c ′ for any t ∈ [ε, π]
. Therefore, we obtain that

I 2 ≤ nσ 2 (r n )πe -c ′ n → 0 .
To deal with I 1 , we use the idea of [GK54, §50, Lemma]: we prove that, for all t ∈ [ 1 rn , ε],

1 -|ϕ λ (t)| ≥ ct 2 σ 2 (1/t) . (6.2)
One could have used Lemma 4 in [START_REF] Mukhin | Local limit theorems for lattice random variables[END_REF], together with Lemmas 1-2 in [START_REF] Mukhin | Local limit theorems for lattice random variables[END_REF] to get that 1 -|ϕ λ (t)| ≥ ct 2 uniformly for t ∈ [0, ε]; this is however not enough for our purpose. Let us introduce P * λ , the distribution of the symmetrised version of P λ , i.e. the law of ξ * = ξ -ξ ′ , where ξ, ξ ′ are two independent copies with law P λ . Then,

|ϕ λ (t)| 2 = ϕ * λ (t) = E * λ [e itξ * ], so we only need to show that 1 -ϕ * λ (t) ≥ ct 2 σ 2 (1/t). We have that 1 -ϕ * λ (t) = R (1 -cos(tx))dP * λ (x) ≥ 1/t -1/t (1 -cos(tx))dP * λ (x) ,
using that 1 -cos(tx) is always non-negative. Now, expanding the cosine (using |tx| ≤ 1), we get

1 -ϕ * (t) ≥ ct 2 1/t -1/t
x 2 dP * λ (x) . Now, let C > 0 be fixed such that c ′ = P(|ξ| ≤ C) > 0; we also have P λ (|ξ| ≤ C) ≥ c ′ 2 for n large enough, since λ → 0. Then, for t small enough so that 1/t ≥ 4C,

1/t -1/t x 2 dP * λ (x) ≥ E ⊗2 λ (ξ -ξ ′ ) 2 1 {|ξ-ξ ′ |≤ 1 t } 1 {|ξ ′ |≤C,|ξ|≥2C} ≥ c ′ 8 E λ ξ 2 1 {2C≤|ξ|≤ 1 t -C} .
Then, since 1/t ≤ r n and λ ≤ c/r n (recall Corollary 5.4), we get that

E λ ξ 2 1 {2C≤|ξ|≤ 1 t -C} ≥ M (λ) -1 e -λrn E ξ 2 1 {2C≤|ξ|≤ 1 t -C} ≥ c ′ σ 2
(1/t) , using also that M (λ) → 1 as λ → 0. This concludes the proof of (6.2).

With (6.2) at hand, we get that

I 1 ≤ nσ 2 (r n ) ε ℓn e -cnt 2 σ 2 (1/t) dt .
Now, for t ∈ (0, 1], let g(t) = tσ 2 (1/t), with σ2 (y) = 2 y 0 uP(|ξ| > u)du which verifies σ(y) ∼ σ 2 (y) as y → ∞ (see (A.5)). Then, g is differentiable and one can easily check that g ′ (t) ≥ 0, at least for t small enough (e.g. using that P(|ξ| > 1/t) = o(t 2 σ 2 (1/t 2 )), see (1.4)). We also let h(t) = t 0 g(u)du, which verifies h(t) ∼ t 2 σ 2 (1/t) as t → 0. Then, by an integration by parts, we get that

ε ℓn e -cnh(t) dt ≤ 1 cng(ℓ n ) e -cnh(ℓn) - ε ℓn g ′ (t) cng(t) 2 e -cnh(t) dt ≤ 1 cng(ℓ n ) e -cnh(ℓn) ,
where the last inequality holds if ε has been fixed small enough so that g

′ (t) ≥ 0 on [0, ε]. Now, using that g(ℓ n ) = ℓ n σ2 (r n ) and h(ℓ n ) ≥ (1 + o(1))ℓ 2 n σ 2 (r n ), I 1 ≤ nσ 2 (r n ) ε ℓn e -cnh(t) dt ≤ 1 ℓ n nσ 2 (r n ) e -cnh(ℓn) ≤ c ′ q * (r n )r n nσ 2 (r n ) e -c ′′ nσ 2 (rn) q * (rn) 2 r 2 n ,
where we have used that ℓ n = c q * (rn)rn ≥ 1 rn for the last inequality. Since r n / nσ 2 (r n ) remains bounded (recall (5.10) and the sentence below), we get that I 1 ≤ c 1 q * (r n )e -c 2 /q * (rn) 2 , which proves that I 1 goes to 0 as n → ∞. Going back to (6.1), we have obtained that

P λ (S n = x n ) ∼ 1 √ 2πnσ 2 (rn)
, which finally gives that

P(S n = x n , M n ≤ ar n ) ∼ 1 2πnσ 2 (r n ) e -nH( xn n ) .
Similarly to what is done for the integral version of the theorem, the case where x n = O(a n ) brings some simplifications: using that σ 2 (r n ) ∼ σ 2 (a n ) and Lemma 2.6, we recover the local limit theorem:

P(S n = x n , M n ≤ ar n ) ∼ 1 √ 2πa n e -x 2 n 2an .

Proof of Proposition 2.16

Recall that we assume that P(ξ = x) is intermediate regularly varying and (2.27) holds, and lim n→∞ xn an = ∞, lim n→∞ xn n = 0. For simplicity, we also assume µ = 0. As in the proof of Proposition 2.8, we focus on the case where lim n→∞ xn n = 0. (We briefly discuss at the end of the proof the case lim inf n→∞ xn n > 0, which can be found e.g. in [START_REF] Armendáriz | Thermodynamic limit for the invariant measures in supercritical zero range processes[END_REF]). We decompose the probability as before, in the following terms:

Q 0 = P(S n = x n , M n > (1 + ε)x n ) , Q 1 = P(S n = x n , (1 -ε)x n ≤ M n ≤ (1 + ε)x n ) , Q 2 = P(S n = x n , εx n < M n < (1 -ε)x n ) Q 3 = P(S n = x n , 4r n < M n ≤ εx n ) , Q 4 = P(S n = x n , r n < M n ≤ 4r n ) .
For the term Q 2 , we have, for some constant c = c(ε) ∈ (0, ∞),

Q 2 ≤ y∈(εxn,(1-ε)xn) nP(ξ = y)P(S n-1 = x n -y, M n-1 ≤ y) ≤ cnP(ξ = x n ) y∈(εxn,(1-ε)xn] P(S n-1 = x n -y) ≤ cnP(ξ = x n )P(S n-1 ≥ εx n )
where we have used Potter's bound (A.2) for the second line. Then, since lim n→∞ xn an = ∞, we have lim n→∞ P(S n-1 ≥ εx n ) = 0, so lim n→∞ Q 2 nP(ξ=xn) = 0. For the term Q 3 , we have (omitting integer parts to lighten notation)

Q 3 = log 2 ( xn rn )-1 k=log 2 ( 1 ε ) P S n = x n , M n ∈ (2 -k-1 x n , 2 -k x n ] ≤ log 2 ( xn rn )-1 k=log 2 ( 1 ε ) y∈(2 -k-1 xn,2 -k xn] nP(ξ = y)P S n-1 = x n -y, M n-1 ≤ 2 k x n ≤ cnP(ξ = x n ) log 2 ( xn rn )-1 k=log 2 ( 1 ε ) 2 ck P S n-1 ≥ 1 2 x n , M n-1 ≤ 2 -k x n ,
where we have used (A.3) and summed over y ∈ (2 -k-1 x n , 2 -k x n ]. We conclude as in the proof of Proposition 2.8 that

Q 3
nP(ξ=xn) ≤ c ′ ε 3 (11/4e) -1/ε . For the term Q 4 , we also get thanks again to (A.3)

Q 4 ≤ y∈(rn,4rn] nP(ξ = y)P S n-1 = x n -y, M n-1 ≤ 4r n ≤ cnP(ξ = x n ) x n r n c P S n-1 ≥ 1 2 x n , M n-1 ≤ 4r n ,
and the conclusion follows exactly as in the proof for the Q 4 of Proposition 2.8. For the term Q 0 , by sub-additivity and using the "almost monotonicity" (2.27), we obtain

Q 0 ≤ n y>(1+ε)xn P(ξ = y)P(S n-1 = x n -y) ≤ cP(ξ = x n )P(S n-1 ≤ -εx n ) , which is negligible compared to P(ξ = x n ), since lim n→∞ xn an = ∞.
It remains only to deal with Q 1 ; we follow Doney's approach [START_REF] Doney | A local limit theorem for moderate deviations[END_REF]. We prove a lower and an upper bound separately. We have We therefore end up with lim ε↓0 lim inf n→∞ Q 1 nP(ξ=xn) ≥ 1. In the other direction, we have

Q 1 ≥ nP(S n = x n , ∃ i s.t. ξ i ∈ ((1 -ε)x n , (1 + ε)x n ), ∀ j = i, ξ j ≤ (1 -ε)x n ) ≥ n (1+ε)xn y=(1-ε)xn P(ξ = y)P(S n-1 = x n -y, M n-1 ≤ (1 -ε)x n ) ≥ n min |y-xn|≤εxn P(ξ = y) P |S n-1 | ≤ εx n , M n-1 ≤ (1 -ε)x n .
Q 1 ≤ n (1+ε)xn y=(1-ε)xn P(ξ = y)P(S n-1 = x n -y) ≤ n max |y-xn|≤εxn P(ξ = y).
Using again the intermediate regular variation of P(ξ = x), we therefore end up with lim ε↓0 lim sup n→∞ Q 1 nP(ξ=xn) ≤ 1. In the case lim inf n→∞ xn n > 0, we can deal with all the terms Q 0 , Q 1 , Q 2 , Q 3 exactly as above. Recalling that in this case we have r n = O(1), its then only remains to show that for any R > 0 the term Q4 := P(S n = x n , M n ≤ R) is negligible compared to nP(ξ = x n ). But one simply bounds Q4 := P(S n = x n , M n ≤ R) ≤ P(S n ≥ x n , M n ≤ R) ≤ e -cxn , by a standard large deviation bound (already mentioned in the integral case); this is negligible compared to nP(ξ = x n ).

Proof of Theorem 2.19

Recall that we assume lim n→∞ xn an = ∞ and that P(ξ = x) is intermediate regularly varying. For simplicity of notation, we also assume that µ = 0, so b n = 0.

We start with the proof of the first convergence, following the approach in [START_REF] Armendáriz | Thermodynamic limit for the invariant measures in supercritical zero range processes[END_REF]. Let 0 < ε < 1. Using (2.31), we know that

lim n→∞ P S n = x n , |M n -x n | ≤ εx n S n = x n , M n > r n = 1 .
For simplicity of notation, let

M 1 = L (ξ) ⊗(n-1) , M 2 = L R(ξ 1 , . . . , ξ n ) S n = x n , |M n -x n | ≤ εx n ,
so we only need to show that lim n→∞ d T V (M 1 , M 2 ) = 0. Let A be any measurable set in Z n-1 . Let 0 < ε < 1. We define the event

B n = {S n -b n = x n } ∩ {M n-1 < (1 -ε)x n } ∩ {|S n-1 -b n | < εx n } , which is a subset of the event {M n-1 < (1 -ε)x n < ξ n = M n < (1 + ε)x n }. Then P(S n = x n , |M n -x n | ≤ εx n )M 2 (A) = P(R(ξ 1 , . . . , ξ n ) ∈ A, S n = x n , |M n -x n | ≤ εx n ) ≥ nP (ξ 1 , . . . , ξ n-1 ) ∈ A, B n , |ξ n -x n | < εx n ≥ n min |t-xn|<εxn P(ξ = t)M 1 (ξ 1 , . . . , ξ n-1 ) ∈ A, M n-1 < (1 -ε)x n , |S n-1 | < εx n .
Using the Potter's bound (A.2) for P(ξ = x), we obtain that min |t-xn|<εxn

P(ξ = t) ≥ c(ε)P(ξ = x n )
for n large enough, with lim ε↓0 c(ε) = 1. We also have

lim n→∞ M 1 (M n-1 < (1 -ε)x n ) = 1, lim n→∞ M 1 (|S n-1 | < εx n ) = 1 ,
using that lim n→∞ xn an = ∞ for the last limit. Thus we get that

M 2 (A) ≥ nP(ξ = x n ) P(S n -b n = x n , |M n -x n | ≤ εx n ) M 1 (A)(1 + o(1)),
where the error term o(1) is uniformly small in all sets A ⊂ Z n-1 . Using Proposition 2.16, we obtain that M 2 (A) ≥ (1 + o(1))M 1 (A) as n → ∞. Since this holds uniformly for any set A, we conclude that lim n→∞ d TV (M 1 , M 2 ) = 0. For the second convergence, letting M2 = L R(ξ 1 , . . . , ξ n ) S n = x n , M n ≤ r n , we also need to prove that d TV (M 1 , M2 ) = 1. But this is clear: if one writes R(ξ 1 , . . . , ξ n ) = (η 1 , . . . , η n-1 ) and consider the event

A n = { n-1 i=1 η i ≥ x n -r n }, then we naturally have that M2 (A n ) = 1, but lim n→∞ M(A n ) = 0, since x ′ n := x n -r n verifies lim n→∞ x ′ n an = ∞.
For the last result, (2.35) follows directly from Proposition 2.16.

7 Law of ξ 1 , . . . , ξ n conditioned on large M n In this section, we prove Proposition 4.1, but first, we give a construction of the regular conditional distribution of R(ξ 1 , . . . , ξ n ) given M n , which is used in the statement (4.2).

A regular conditional distribution

We give here a concrete construction of L R(ξ 1 , . . . , ξ n ) M n = x , x ∈ R (7.1) as follows:

• If P(x -δ ≤ ξ ≤ x + δ) = 0 for some δ > 0, i.e. x is not in the support, we can basically assume any conditional distribution. But for simplicity, let

L R(ξ 1 , . . . , ξ n ) M n = x = L ξ ⊗(n-1) . (7.2)
• If P(ξ = x) > 0, then we use the usual conditional probability formula to define (7.1).

• If P(ξ = x) = 0 and P(x -δ ≤ ξ ≤ x + δ) > 0 for any δ > 0, it turns out that lim

δ→0 d TV L R(ξ 1 , . . . , ξ n ) x -δ ≤ M n ≤ x + δ , L ξ ξ ≤ x ⊗(n-1) = 0. (7.3)
This is proved in Proposition 7.1 below. Then we can define

L R(ξ 1 , . . . , ξ n ) M n = x = L ξ ξ ≤ x ⊗(n-1) . (7.4)
Using standard measure-theoretic approaches, the construction indeed gives a version of regular conditional distribution. It remains to prove the following proposition.

Proposition 7.1. Let ξ be any real-valued random variable. Assume that x ∈ R is such that P(ξ = x) = 0 and P(x -δ ≤ ξ ≤ x + δ) > 0 for any δ > 0, then (7.3) holds.

Proof. Fix n ∈ N and x ∈ R satisfying the assumptions of Proposition 7.1. First of all, by assumption, it is straightforward to see that since F is continuous at x,

P M n ∈ [x -δ, x + δ] ∼ nP ξ ∈ [x -δ, x + δ] F (x) n-1 = o(1), as δ ↓ 0. (7.5) For A ⊂ B(R n-1 ), let us bound P R(ξ 1 , . . . , ξ n ) ∈ A M n ∈ [x -δ, x + δ] -L ξ ξ ≤ x ⊗(n-1) (A) = P R(ξ 1 , . . . , ξ n ) ∈ A, M n ∈ [x -δ, x + δ] P(M n ∈ [x -δ, x + δ]) -L ξ ξ ≤ x ⊗(n-1) (A)
by the sum of three terms P 1 , P 2 , P 3 that we now define and treat separately. The first term is

P 1 := P(∃ i = j ≤ n : ξ i , ξ j ∈ [x -δ, x + δ]) P(M n ∈ [x -δ, x + δ]) ≤ n 2 P(ξ ∈ [x -δ, x + δ]) 2 P(M n ∈ [x -δ, x + δ]) ---→ δ→0 0 ,
where we have used (7.5) and the assumption that P(ξ = 0) to obtain the last limit. The second term P 2 is

P R(ξ 1 , . . . , ξ n ) ∈ A ∩ (-∞, x -δ) n-1 , M n ∈ [x -δ, x + δ] P(M n ∈ [x -δ, x + δ]) -L ξ ξ < x -δ ⊗(n-1) (A) = nP(ξ ∈ [x -δ, x + δ])F (x -δ) n-1 P(M n ∈ [x -δ, x + δ]) -1 L ξ ξ < x -δ ⊗(n-1) (A) ---→ δ→0 0 ,
where we have used (7.5) to obtain the last limit. The last term is

P 3 := L ξ ξ ≤ x ⊗(n-1) (A) -L (ξ ξ < x -δ ⊗(n-1) (A) ≤ nP ξ ∈ [x -δ, x] ξ ≤ x ---→ δ→0 0 ,
where the last limit is due to the fact that P(ξ = x) = 0. This concludes the proof of (7.3).

Proof of Proposition 4.1

Let us first prove (4.2). For A ⊂ B(R n-1 ), we bound

P R(ξ 1 , . . . , ξ n ) ∈ A | M n ≥ x n -P (ξ 1 , . . . , ξ n-1 ) ∈ A = P R(ξ 1 , . . . , ξ n ) ∈ A, M n ≥ x n P(M n ≥ x n ) -P (ξ 1 , . . . , ξ n-1 ) ∈ A
by the sum of three terms P 1 , P 2 , P 3 that we now define and treat separately. The first term is

P 1 := P(∃ i = j ≤ n : ξ i , ξ j ≥ x n ) P(M n ≥ x n ) ≤ n 2 P(ξ ≥ x n ) 2 P(M n ≥ x n ) ---→ n→∞ 0 , since we have P(M n ≥ x n ) ∼ nP(ξ ≥ x n ) = o(1) as n → ∞ (7.6)
by assumption. The second term P 2 is

P R(ξ 1 , . . . , ξ n ) ∈ A ∩ (-∞, x n ) n-1 , M n ≥ x n P(M n ≥ x n ) -P (ξ 1 , . . . , ξ n-1 ) ∈ A ∩ (-∞, x n ) n-1 = nP(ξ ≥ x n )F (x n -) n-1 P(M n ≥ x n ) -1 P (ξ 1 , . . . , ξ n-1 ) ∈ A ∩ (-∞, x n ) n-1 ---→ n→∞ 0 ,
where the last limit comes from (7.6). The last term is P 3 := P(M n-1 ≥ x n ), which also goes to 0 as n → ∞, thanks to (7.6).

Now we turn to the proof of (4.3). By assumption, for the sequence (x n ) n≥1 , we know that

P(M n-1 ≥ x n ) ∼ (n -1)F (x n ) = o(1) as n → ∞. Then d TV L ξ ξ ≤ x n ⊗(n-1) , L ξ ⊗(n-1) -→ 0, as n → ∞. (7.7)
There are two cases we need to treat: (i) (x n ) n≥1 is a sequence such that P(ξ = x n ) = 0 for any n; (ii) (x n ) n≥0 is a sequence such that P(ξ = x n ) > 0 for any n. For case (i), by (7.2), (7.4) and (7.7), we conclude that (4.3) holds. Let us now consider case (ii). Note that nF (

x n ) = o(1) implies that nP(ξ = x n ) = o(1) as n → ∞. Then P(M n = x n ) ∼ nP(ξ = x n ) = o(1) as n → ∞. (7.8) For A ⊂ B(R n-1 ), we again bound P R(ξ 1 , . . . , ξ n ) ∈ A | M n = x n -P (ξ 1 , . . . , ξ n-1 ) ∈ A = P R(ξ 1 , . . . , ξ n ) ∈ A, M n = x n P(M n = x n ) -P (ξ 1 , . . . , ξ n-1 ) ∈ A
by the sum of three terms P1 , P2 , P3 , that we now define and treat separately. The first term is P1 :=

P(∃ i = j ≤ n : ξ i , ξ j = x n ) P(M n = x n ) ≤ n 2 P(ξ = x n ) 2 P(M n = x n ) ---→ n→∞ 0 , using (7.8). The second term P2 is P R(ξ 1 , . . . , ξ n ) ∈ A ∩ (-∞, x n ) n-1 , M n = x n P(M n = x n ) -P (ξ 1 , . . . , ξ n-1 ) ∈ A ∩ (-∞, x n ) n-1 = nP(ξ = x n ) P(M n = x n ) -1 P (ξ 1 , . . . , ξ n-1 ) ∈ A ∩ (-∞, x n ) n-1 ---→ n→∞ 0 ,
using again (7.8). The last term is P3 := P(M n-1 ≥ x n ), which goes to 0, thanks to (7.6). This concludes the proof of (4.3).

A Discussions on Rozovskii's theorem

In this section, we make some comments on Rozovskii's result, in several directions:

• We discuss the condition under which Theorem 2.1 is true: Rozovskii states its result in terms of the function q(•), recall the definition (2.1); we show in Section A.1 that it is equivalent to (2.2), which is in turn equivalent to F being extended regularly varying.

• We discuss how one may find equivalent (or sufficient) conditions to have (2.4): our goal is to obtain a more tractable condition in order to get the large deviation asymptotics (2.3); this is the purpose of Proposition A.2 below.

• We provide some examples of distributions for which Rozovskii's result can be applied, some for which it cannot be applied.

Again, for simplicity of notation, we assume that µ = E[ξ] = 0 in this appendix.

A.1 About the condition (2.2) and extended/intermediate regular variation

In [Roz90, Thm. 6], Rozovskii assumes that the function q defined in (2.1) by q(x) = x 2 F (x)/σ 2 (x) verifies that there exists some c > 0 such that x c q(x) is asymptotically equivalent to a non-decreasing function. We now state some equivalent statements and in particular we show that it is equivalent to F being extended regularly varying, see (2.10).

Claim A.1. Assume that x → σ 2 (x) is slowly varying at ∞. The following statements are equivalent:

(i) ∃ c > 0 such that x c q(x) is equivalent to a non-decreasing function;

(ii) ∃ c > 0 such that x c F (x) is equivalent to a non-decreasing function, i.e. (2.2);

(iii) there is a constant c > 0 and a function y → δ y such that lim y→∞ δ y = 0, for which one has

F (x) ≤ F (y) ≤ (1 + δ y ) x y c F (x) ∀x ≥ y ; (A.1) (iv)
The function F is extended regularly varying at infinity (see (2.10)).

Proof. (i) ⇒ (ii). If x c q(x) is asymptotically equivalent to a non-decreasing function v(x), then

x c+2 F (x) = x c σ 2 (x)q(x) = (1 + o(1))σ 2 (x)v(x),
with both σ 2 (x) and v(x) non-decreasing. Thus condition (2.2) follows. (ii) ⇒ (i). If x c F (x) is asymptotically equivalent to a non-decreasing function v(x) (i.e., assuming (2.2)), then x c q(x) = (1+o(1))v(x)x 2 /σ 2 (x). Note that x 2 /σ 2 (x) is asymptotically equivalent to x 2 /σ 2 (x) (recall that σ(x)

:= E[(|ξ|∧x) 2 ]) which is differentiable with derivative 2x σ 2 (x) (1-x 2 P(|ξ|>x)
σ 2 (x) ) ≥ 0 for x large. Hence x c q(x) is asymptotic to a non-decreasing function. (ii) ⇒ (iii). If x c F (x) is asymptotically equivalent to an increasing function v(x), we can write x c F (x) = (1 + ε x )v(x), with lim x→∞ ε x = 0. Hence, for y ≤ x, using that v is increasing, we have

y c F (y) x c F (x) = 1 + ε y 1 + ε x v(y) v(x) ≤ 1 + |ε y | 1 -inf x≥y |ε x | → 1, as y → ∞,
which proves the upper bound in (A.1) (the lower bound is trivial).

(iii) ⇒ (ii). Let v(y) = y c F (y) and ṽ(y) = inf x≥y v(x). Then, clearly, ṽ is non-decreasing and by (A.1) we have that (1 + δ y ) -1 v(y) ≤ ṽ(y) ≤ v(y), so v(y) and ṽ(y) are asymptotically equivalent. Hence (ii) is verified.

(iii) ⇒ (iv) is obvious from the definition of extended regular variation; note that (A.1) gives the upper and lower Matuszewska indices are c and 1 respectively.

A.2.1 Preliminary observations

Let us first give some consequences of the intermediate regular variation of F , which we recall is equivalent to (A.2).

Claim A.4. Assume that x → σ 2 (x) is slowly varying at infinity and that F is intermediate regularly varying. Let x ≥ y > 0. Then for y large enough

y x 2 ≤ q(y) q(x) ≤ (1 + δ y )κ x y y x , (A.7)
where δ y and κ(•) are the same as in (A.2).

Proof. Note that by (A.2), we have 1 ≤ F (y) F (x) ≤ (1 + δ y )κ( x y ) for all x ≥ y > 0. Then we can use the definition (2.1) of q to obtain σ 2 (x) σ 2 (y)

y x 2 ≤ q(y) q(x) ≤ (1 + δ y )κ x y σ 2 (x) σ 2 (y) y x 2 .
Note that σ 2 (x) ≥ σ 2 (y) for x ≥ y > 0 using monotonicity, and σ 2 (x) 2 (y) ≤ x y for y large enough using Potter's bound for slowly varying function. Then we obtain (A.7).

Lemma A.5. Assume that x → σ 2 (x) is slowly varying at infinity and that F is intermediate regularly varying. Then uniformly for x C| log q(x)| ≤ y ≤ x, we have

| log q(x)| ∼ | log q(y)| as x → ∞.
In particular, recalling that ω n :=

an √ | log q(an)| , we have | log q(ω n )| ∼ | log q(a n )| as n → ∞.
Similarly, the above display also holds uniformly for x ≤ y ≤ Cx| log q(x)|.

Proof. We will prove only the first part as the second part follows similar lines of reasoning.

Recall that lim x→∞ q(x) = 0, see (1.4). Let us assume that x is large enough so that log q(x) < 0 and let us set h := x/y, which verifies 1 ≤ h ≤ C| log q(x)|. Using the second inequality in (A.7), we have

q(y) q(x) ≤ (1 + δ y ) κ(h) h ≤ 2 κ(h) h , for x large enough.
Then log q(x) ≥ log q(y) -log 2κ(h) + log h, so that

| log q(x)| ≤ | log q(y)| + log 2κ(h) ≤ | log q(y)| + log 2κ(C| log q(x)|) ,
using that κ is non-decreasing. On the other hand, the first inequality in (A.7) shows that q(y) q(x) ≥ h -2 , which implies

| log q(x)| ≥ | log q(y)| -2 log h ≥ | log q(y)| -2 log κ(C| log q(x)|).
Since lim x→∞ q(x) = 0 and recalling that lim s→∞ 1 s log κ(s) = 0, see (A.2), this concludes the proof. by (A.15) and σ 2 (x)

| log q(x)| ∼ σ 2 (y) | log q(y)| . Indeed the later holds due to (A.13) and Lemma A.5. Moreover, we can replace x by 2x in the above display: indeed, we have σ 2 (2x) ∼ σ 2 (x) (since σ 2 (•) is slowly varying) and | log q(2x)| ∼ | log q(x)|, see Lemma A.5. Remark A.9. Assume that Proposition A.3, i.e. (A.6), holds. If x n ∼ ca n | log q(a n )| for some constant c > 0, then recalling the asymptotics (5.18) of r n and (A.13), we have r n ∼ an c √ | log q(an)| and σ 2 (r n ) ∼ σ 2 (a n ) (using also that σ 2 (•) is slowly varying). Therefore, we have σ 2 (r n ) ∼ σ 2 (a n ) uniformly for a n ≤ x n ≤ ca n | log q(a n )|, using that σ 2 (•) is an increasing function (we also clearly have r n ∼ a n /c and so σ 2 (r n ) ∼ σ 2 (a n ), if x n ∼ ca n ).

A.3 A few examples

First, let us give a generic example where Rozovskii's condition is verified.

Example A.10. Assume that F (x) ∼ L(x)x -β as x → ∞, for some slowly varying function L(•) and some β ≥ 2. Assume also that the left tail is "dominated" by the right tail, in the sense that F (-x) ∼ cF (x) for some c ≥ 0; with F (-x) = o(F (x)) if c = 0. This is for instance the case if ξ = X -E[X] for some non-negative random variable X.

If β > 2, then we have E[|ξ| 2+δ ] < ∞ for some δ > 0, so Nagaev's condition is verified. We will therefore focus on the case β = 2: Nagaev's condition is not verified, but we will see that Rozovskii's condition (2.4) is verified for a n defined by a 2 n = nσ 2 (a n ). If β = 2, then we have σ 2 (x) = 2 x 0 tP(|ξ| > t)dt, with P(|ξ| > t) ∼ (1 + c)L(t)t -2 as t → ∞. Denote ℓ(t) = 2t 2 P(|ξ| > t), so that σ(x) 2 = x 0 ℓ(t)t -1 dt (A.16) and note that ℓ(x) ∼ 2(1 + c)L(x) as x → ∞. Then we have that q(x) ∼ c ′ ℓ(x)/σ 2 (x), with ℓ(x)/σ 2 (x) which is known to verify lim x→∞ ℓ(x)/σ 2 (x) = 0, see [BGT89, Prop. 1.5.9.a]. Then, thanks to Proposition A.3, we simply need to verify that σ 2 (x | log q(x)|) -σ 2 (x) = o σ 2 (x) | log q(x)| . (A.17) Since σ 2 (x)-σ 2 (x) = x 2 P(|ξ| > x) = ℓ(x) and since ℓ(x) ∼ 2(1+c)σ 2 (x)q(x) = o σ 2 (x) | log q(x)| , it suffices to prove the above display with σ 2 (•) replaced by σ 2 (•) on the l.h.s. of (A.17), thanks to Lemma A.5. We can use Claim A.11 below to get that 0 ≤ σ 2 (x | log q(x)|) -σ 2 (x) ≤ ℓ(x) 1 2 log | log q(x)| + ε x | log q(x)| ≤ Cσ 2 (x)q(x) | log q(x)| = o σ 2 (x) | log q(x)| , which concludes the proof.

Claim A.11. Let ℓ be a non-negative slowly varying function. Then there exists ε x that verifies lim x→∞ ε x = 0 such that for any z ≥ Now, using that g is regularly varying of index -1, we have that uniformly for u ≥ 1, g(ux)/g(x) = u -1 + ε x with lim x→∞ ε x = 0 (this is the uniform convergence theorem, see [START_REF] Nicholas | Regular variation[END_REF]). We end up with 0 ≤ z x ℓ(t)t -1 dt ≤ ℓ(x) log z x + ε x z x -1 , which concludes the proof.

We present an example where the left tail is heavier than the right tail: we give some conditions for (2.4) to hold; this will also stress the importance of the choice of the normalising sequence (a n ) n≥1 .

Example A.12. Assume that ξ is centered and that σ 2 = Var(ξ) < ∞. We will focus on a case where Nagaev's condition E[|ξ| 2+δ ] < ∞ for some δ > 0 is not satisfied. We consider a case where | log q(x)| ≍ log x as x → ∞; this is ensured for instance if the right tail verifies x -β + ≤ F (x) ≤ x -β -for some β + > β -> 2 (for x large).

Our goal is to discuss the conditions imposed by (2.4) on the left tail of the distribution. We assume that F (-x) ∼ ℓ(x)x -2 for some slowly varying function ℓ(•); think about taking ℓ(x) = (log x) -a for some a > 1 (so that σ 2 < ∞). We now compare two possible choices for the normalising sequence (a n ) n≥1 .

If one defines a n = σ √ n, then recalling (2.7), (A.5) and (A.16), (2.4) is equivalent to

∞ x ℓ(t) t dt ∼ E[ξ 2 1 {|ξ|>x} ] = o 1 log x ,
(recall also that | log q(x)| ≍ log x). Since ℓ(x) = (log x) -a for some a > 1, this amounts to having a > 2 for this particular choice of a n . If on the other hand one defines a n by a 2 n = nσ(a n ), then Proposition A.3 tells that (2.4) is equivalent to

E[ξ 2 1 {x<|ξ|<x √ | log q(x)|} ] ∼ x √ | log q(x)| x ℓ(t) t dt = o 1 log x
By Claim A.11, we have that the integral is o(ℓ(x) √ log x) (using that | log q(x)| ≍ log x), so it is enough to have ℓ(x) ≤ (log x) -3/2 (or a > 3/2). If ℓ(x) is non-increasing, then the integral is bounded by a constant times ℓ(x) log log x, so a sufficient condition becomes ℓ(x) = o( 1 log x log log x ); this is verified if ℓ(x) = (log x) -a with any a > 1. Example A.13. Let us consider the case where ξ is centered and where the left and right tails verify F (x) ∼ L(x)x -2 and F (-x) ∼ ℓ(x)x -2 , with the left tail being heavier, that is L(x) = o(ℓ(x)) as x → ∞. In this case, q(x) is slowly varying and the interplay between the slowly varying functions L, ℓ, σ 2 is subtle. For simplicity, let us consider the case where ℓ(x) ∼ (log x) -a for some a ∈ R. We will find conditions on the right tail for (2.4) to be verified.

If σ 2 < ∞ (i.e. a > 1), let us set for now a n = σ √ n. We have that

E[ξ 2 1 {|ξ|>x} ] ∼ ∞ x
ℓ(t) t dt ∼ (log x) 1-a , so (2.7) is equivalent to | log q(x)| = o((log x) a-1 ). If a ≥ 2, the fact that q is slowly varying ensures that | log q(x)| = o(log x), thus (2.7) holds. If a ∈ (1, 2), we have to check the right tail: for instance, if L(x) ≍ e (log x) b for some b ∈ (0, 1), then | log q(x)| ≍ (log x) b and the condition (2.7) is verified provided that a -1 > b.

If we define a n by the relation a 2 n = nσ 2 (a n ), we to consider

E ξ 2 1 {x<|ξ|<x √ | log q(x)|} ∼ x √ | log q(x)| x ℓ(t) t dt ∼ 1 2 (log x) -a log | log q(x)| .
Therefore, (2.6) is equivalent to | log q(x)| log(| log q(x)|) = o σ 2 (x)(log x) a . If a > 1, this is automatic, since we have | log q(x)| = o(log x) as noticed above. If a = 1, this is also verified thanks to the same observation, using also that σ 2 (x) ∼ log log x. If a < 1, then σ 2 (x) ≍ (log x) 1-a , so the condition becomes | log q(x)| log | log q(x)| = o(log x); this is verified for instance if in the right tail we have L(x) ≍ e (log x) b for some b ∈ (0, 1), but is not verified if one takes for instance L(x) ≍ exp(log x/ log log x).

We now provide an example of a distribution in the domain of attraction of the normal law where one cannot find any normalising sequence (a n ) n≥1 that verifies Rozovskii's condition (2.4).

Example A.14. The example is constructed to make the left tail too heavy compared to the right tail. Take some function q(x) and suppose that σ 2 (x) = exp log x/ log (3) q(x) , for x large enough with log (3) t = log log log 1 t . Let (a n ) n≥1 be any normalising sequence, i.e. verifying (1.5).

Since ω n = a n / | log q(a n )|, we have that log ω n log (3) q(ω n ) = log a n log (3) q(ω n ) -1 2 log | log q(a n )| log (3) q(ω n ) .

So if we assume that log (3) q(ω n ) -log (3) q(a n ) = o(log (3) q(a n )/ log a n ) (take for instance q(y) = y -1 or q(y) = 1/ log y), we get that log ω n log (3) q(ω n ) = log a n log (3) q(a n ) -1 2 log | log q(a n )| log (3) q(a n ) + o(1) .

Hence B One-big-jump phenomenon when α ∈ (0, 2)

σ 2 (ω n ) ∼ e
In this section we prove Corollaries 3.2 and 3.5. We begin with the following lemma.

Lemma B.1. For any ξ that is non-degenerate (i.e. whose law is not concentrated on a single point), we have that Hence (B.1) holds in this case as well.

B.1 Proof of Corollary 3.2

The statement (3.1) is already contained in the proof of [Ber19a, Theorem 2.1], so we focus on the proof of (3.2). The fact that lim n→∞ xn an = ∞ is sufficient to obtain (3.2) can be seen by using (3.1) and the statement (4.2) in Proposition 4.1. We will prove here the necessity.

Let (x n ) n≥1 be such that Next we assume l ∈ (-∞, ∞) and we prove that (3.2) does not hold. We split the proof according to several cases: recall our assumption (1.2), which ensures the convergence of S n (properly centered and normalised) to an α-stable random variable S α , recall (1.1). (i) If α ∈ (0, 1), p = 1, and l ∈ (-∞, 0]. The limiting α-stable random variable S α is then supported on [0, ∞) (see [START_REF] Feller | An introduction to probability theory and its applications[END_REF]XIII.7 Thm. 2] and the following remark). Then we

. 9 )

 9 Similar results hold if we replace {Sn -b n ≥ x n } in the conditioning by {S n -⌊b n ⌋ = x n }, assuming P(ξ = x)is intermediate regularly varying and (2.27), and replacing the condition on nF (xn) P(Sn-bn≥xn) by a condition on nP(ξ=xn)

y 2 ,

 2 with µ(y) = E[ξ1 |ξ|≤y ]. Since E[ξ] = 0, we have µ(y) = -E[ξ1 |ξ|>y ] ≤ E[|ξ|1 |ξ|>y ]. Therefore, thanks to Claim 5.1 above, we have that E[|ξ|1 |ξ|>y ] = o(σ 2 (y)/y): if y is large enough we have yµ(y) ≤ σ 2 (y), so nµ(y) y -nσ 2 (y) y 2

e

  ux 1 (-∞,r] (x) , with M (u) = M r (u) := E[e uξ 1 {ξ≤r} ] .

  .5) Now, recalling the definition (2.13) for r n , we get that lim n→∞ r n = ∞ since lim n→∞ xn n = 0. Thanks to Claim 5.1, we therefore get that |m(0)| = o(r -1 n σ 2 (r n )). In particular, in view of (2.13), we get that m(0) = o( xn n ). From Claim 5.3, if u = s rn then m(u) = M ′ (u) M (u) = (1 + o(1)) s rn σ 2 (r n ) = (1 + o(1))s xn n (using that σ 2 is slowly varying), with the o(1) uniform for all s ∈ [c, d]. Hence, if t = s xn n we get that λ(t) = (1 + o(1)) s rn with the o(1) uniform for s ∈ [c, d].

x 2 n 2a 2 n

 2 + o(1), at least for an ≤ xn ≤ Can | log q(an)|, one needs (2.2) and Rozovskii's condition (2.4), see Remark 2.7.

  Since lim n→∞ xn an = ∞, the last probability in the last line goes to 1. Using also that P(ξ = x) is intermediate regularly varying, recall (2.11), we have that lim ε↓0 lim inf n→∞ min |y-xn|≤εxn P(ξ = y) P(ξ = x n ) = 1 .

  t dt ≤ ℓ(x) log z x + ε x z x -1 , as x → ∞ .Proof. Setting g(y) := ℓ(y)y -1 , we get by a change of variable that z x ℓ(t)t -1 dt = ℓ(x)

σ 2

 2 q(an)| log (3) q(an) nσ 2 (a n ) (ω n ) -a 2 n ω 2 n → ∞,which tells that (2.4) does not hold.

d

  TV L R(ξ 1 , . . . , ξ n ) , L (ξ) ⊗(n-1) does not converge to 0 as n → ∞.(B.1)Proof. Let us denote (η 1 , η 2 , . . . , η n-1 ) = R(ξ 1 , ξ 2 , . . . , ξ n )and N n be the second largest variable in ξ 1 , . . . , ξ n , so N n = max{η 1 , . . . , η n-1 }. Define T := sup{t : P(ξ > t) > 0} and let us distinguish two cases.If P(ξ = T ) > 0 it is easy to see thatP(η 1 = T ) = P(ξ 1 = ξ 2 = T ) = P(ξ = T ) 2 < P(ξ 1 = T ).Thus (B.1) holds in this case.If P(ξ = T ) = 0, let x converge to T from the left and put n = n(x) = ⌊ 1 P(ξ>x) ⌋. Thenlim n→∞ P(M n-1 ≤ x) = lim n→∞ (1 -P(ξ > x)) n-1 = e -1 ,andlim n→∞ P(N n ≤ x) = lim n→∞ P(M n ≤ x) + lim n→∞ nP(ξ > x)P(M n-1 ≤ x) = 2e -1 .Therefore lim n→∞ P(N n ≤ x) = e -1 = 2e -1 = lim n→∞ P(M n-1 ≤ x).

d

  TV L R(ξ 1 , . . . , ξ n ) S n -b n ≥ x n , L R(ξ 1 , . . . , ξ n ) = 0, (B.3) since lim n→∞ P(S n -b n ≥ x n ) = 1. By Lemma B.1, (3.2) does not hold.

Recall that a distribution is called subexponential if for any y ∈ R we have limx→∞ F (x + y)/F (x) = 1 and P(S2 ≥ x) ∼

2F (x) as x → ∞. See for example[START_REF] Foss | An introduction to heavytailed and subexponential distributions[END_REF] for further information and background.

A function f is O-regularly varying if -∞ < lim infx→∞ f (λx) f (x) ≤ lim sup x→∞ f (λx) f (x) < +∞ for all λ ≥ 1.

Let us mention that there is a notational inaccuracy in[START_REF] Armendáriz | Zero-range condensation at criticality[END_REF]: the authors in fact assume that w(n) ∼ cn -b for some constant c > 0, but for this, one would need the additional condition k≥1 ε k k < ∞.

Acknowledgement

The authors thank Jochen Blath, Denis Denisov, Sergey Foss, Dalia Terhesiu and Vitali Wachtel for their friendly comments and pointers to the literature.

(iv) ⇒ (iii) follows by Potter's bound for extended regularly varying functions, see [Cli94, Thm. 2.3]: for any δ > 0, there is some x 0 > 0 such that for any λ < 1 and x ≥ x 0 1 ≤ F (λx) F (x) ≤ (1 + δ)λ -c-1 , where c is the upper Matuszewska index; the lower bound simply follows from the fact that F is non-increasing.

Claim A.2. The fact that function F is intermediate regularly varying (see (2.11)) is equivalent to the following: there exists some non-decreasing function κ

with κ satisfying lim s→1+ κ(s) = 1 and lim s→∞ 1 s log κ(s) = 0. Additionally, there is a constant c > 0 such that κ(s) ≤ cs c , for all s ≥ 1. (A.3)

Proof. Note that (A.2) readily implies intermediate regular variation by verifying the definition. For the reverse implication, we proceed exactly as for Claim A.1: using Potter's bound [Cli94, Thm. 2.3] for intermediate regularly varying functions, we obtain that for all δ > 0 there is some x 0 > 0 such that for any λ > 1 and

The fact that κ grows at most polynomially follows directly from the representation theorem for intermediate regularly varying functions, see [START_REF] Daren | Intermediate regular and Π variation[END_REF]Cor. 3

.2I].

A.2 About the condition (2.4) in Theorem 2.1

We now find equivalent conditions to the condition (2.4) in Theorem 2.1. Recall that we defined

and let (a n ) n≥1 be a sequence defined by the relation (2.5). Then, noticing that

we have lim y→∞

= 0 thanks to (1.4), so

This shows that (a n ) n≥1 defined by (2.5), i.e. a 2 n = nσ(a n ), is a valid normalising sequence since it verifies (1.5) (with α = 2).

Proposition A.3. Let α = 2, let (a n ) n≥1 be defined by (2.5), and assume that F is intermediate regularly varying. Then the relation (2.4) is equivalent to (2.6), that is to

The rest of Section A.2 will be devoted to the proof of this proposition.

Remark A.6. Thanks to Lemma A.5, if x n ∼ ca n | log q(a n )| for some c > 0, we get that

On the other hand, recalling the definition (1.5) of a n ,

where we used Potter's bound since σ(•) is slowly varying. In conclusion, a sufficient condition to have that

A.2.2 Proof of Proposition A.3

Recall that (a n ) n≥1 is defined by (2.5) and that we assumed that F is intermediate regularly varying. Let us recall the definition

1).

Lemma A.7. The condition (2.4) is equivalent to

The above conditions (A.8)-(A.9) are further equivalent to

Using the definition (2.5) of a n , we can write σ 2 (an) | log q(an)| = n -1 ω 2 n , so we obtain that (2.4) is equivalent to (A.8) and (A.9). Next, we show that (A.10) is equivalent to (A.8)-(A.9).

Step 1. We first show that (A.10) implies (A.8); we actually prove that (A.10) implies

Note that we have

If (A.10) holds, then we have

Using Lemma A.5, we therefore get that

Then using (A.12), we obtain lim n→∞ nP(ω n < |ξ| ≤ a n ) = 0.

It remains to show that lim n→∞ nP(|ξ| > a n ) = 0. But this is simply due to the fact that n = a 2 n σ 2 (an) (see definition (2.5)) and (1.4). Thus we have shown that (A.10) implies (A.8).

Step 2. We now show that (A.10) implies (A.9). Thanks to (A.4) and (A.14), we only have to show that

Recall that (A.10) implies (A.11). Using that

| log q(an)| as n → ∞, which is further equivalent to the following, using (A.14),

This concludes the proof since ω n can be replaced by any sequence growing to infinity (one can use the argument that ω ⌊cn⌋ ∼ √ cω n for any fixed c > 0, which can be proved using Lemma A.5 and the fact that a n also satisfies this property).

Step 3. Finally we show that (A.9) implies (A.10). Note that (A.9) implies (A.13). Applying Lemma A.5, we find that (A.14) holds. We also observe that

where the last step is due to (A.9) and (A.13). This completes the proof, thanks to (A.14).

Proof of Proposition A.3. By Lemma A.7, setting y := x/ | log q(x)|, the condition (2.4) is equivalent to

This is because (A.10) still holds if we replace (a n ) n≥1 by an arbitrary increasing sequence. We only need to show that (A.15) is equivalent to (A.6). To this purpose, it suffices to show that

under either the assumption of (A.15) or (A.6). This is implied by the following claim, using that y | log q(y)| ∼ x thanks to Lemma A.5 (and the fact that σ 2 is non-decreasing).

Claim A.8.

Proof. Assume that (A.15) holds; the proof is identical if we assume that (2.6) holds instead. First, since y = x/ | log q(x)|, we have that y ≤ 1 2 x for x large. Hence,

have that lim n→∞ P(S n -b n > x n ) = 1 and we deduce (B.3). By the same argument as in the case l = -∞, we conclude that (3.2) does not hold.

(ii) In the rest, we treat all the following cases in the same way (so we will not distinguish them in the following analysis):

Let Λ be the Lévy measure associated to the α-stable random variable S α , given by Λ(dx) =

2-α

α |x| -α-1 (p1 {x>0} + q1 {x<0} )dx; the characteristic function of S α is given by:

and lim

where Λ y = Λ([y, ∞)). To prove that (3.2) does not hold, it suffices to show that there exists some y > 0 such that lim n→∞ P(a

, or in other words that there is some y > 0 such that

Considering jumps that occur on [0, 1], let Y

(1) 1

be the sum of all jump sizes that are not smaller than y; let also

and Y

(2) 1 are independent, with respective characteristic functions

(1) 1 = 0: using this fact, we obtain

where K denotes the number of jumps with size not smaller than y. Note that K follows the Poisson distribution Poi(Λ y ) with parameter Λ y and is independent of Y

(2)

1 . Then we have

where we have used the fact that V ≥ y. Note that, as y → ∞, we have that Y

(2) 1 converges in distribution to Y 1 and P(Y

(2) 1 ≥ l -y) goes to 1. Since Λ y → 0, as y → ∞, we use the above display to obtain that, as y → ∞,

Therefore we have that P(K > 0)P(Y

≥ l -y) < 0 for y large enough. Going back to (B.9), we end up with

for y large enough, which proves (B.8). This completes the proof for (3.2).

B.2 Proof of Corollary 3.5

In the following, to simplify notation, we assume that b n = 0. The proof that lim n→∞ xn an = ∞ is a sufficient condition to get (3.6) follows the same line of proof as that of Theorem 2.19, see Section 6.3 (following ideas of [START_REF] Armendáriz | Thermodynamic limit for the invariant measures in supercritical zero range processes[END_REF]), so we skip it.

We now show that lim n→∞ xn an = ∞ is necessary. Assume that that (B.2) holds, i.e. that lim n→∞ xn an = l ∈ [-∞, ∞). We will distinguish two set of cases (i) and (ii). (i) Consider first the cases

(B.10) If (3.6) holds, due to (1.1), we have that

Note that l -S α < 0 for the first case in (B.10) (recall that S α is supported on [0, ∞)) and that l -S α = -∞ in the other cases. Notice also that bn nan and xn nan both go to 0 as n → ∞ (indeed b n = 0 if α ∈ (0, 1) and b

slowly varying): the above display therefore implies that

But we clearly have that P(M n < 1 n (b n +x n ), S n = b n +x n ) = 0, which brings a contradiction. We conclude that (3.6) is not true for the cases in (B.10).

(ii) Next we consider the cases in (B.4). Let g α be the density function of S α : by Stone's local limit theorem [START_REF] Stone | On local and ratio limit theorems[END_REF], we have P(S n -b n = x n ) = 1 a n g α (l) + o(1) , as n → ∞.

Here g α (l) > 0 for any l in the ranges given in (B.4) in different cases. Choose any ε ∈ (0, 1) and any t ∈ R such that P(ξ > t) > ε. Then P(M n ≤ t) < (1 -ε) n . Combining with the above, we get that

which goes to 0 as n goes to ∞, since a n is regularly varying. We therefore obtain that lim n→∞ P(M n > t | S n -b n = x n ) = 1, which in turn implies that which is a contradiction. This proves that (3.6) is not true in cases (B.4). This concludes the proof.