Yuri Goegebeur
email: yuri.goegebeur@imada.sdu.dk

Armelle Guillou
email: armelle.guillou@math.unistra.fr

Gilles Stupfler
email: gilles.stupfler@univ-amu.fr

Uniform asymptotic properties of a nonparametric regression estimator of conditional tails Propriétés asymptotiques uniformes d'un estimateur non-paramétrique de l'indice des valeurs extrêmes conditionnel

Keywords: Tail-index, kernel estimation, strong uniform consistency

We consider a nonparametric regression estimator of conditional tails introduced by Goegebeur, Y., Guillou, A., Schorgen, G. (2013). Nonparametric regression estimation of conditional tails -the random covariate case. It is shown that this estimator is uniformly strongly consistent on compact sets and its rate of convergence is given.

Résumé. Nous considérons l'estimateur à noyau de l'indice des valeurs extrêmes conditionnel présenté dans Goegebeur, Y., Guillou, A., Schorgen, G. (2013). Nonparametric regression estimation of conditional tails -the random covariate case. Nous montrons la consistance uniforme presque sûre de cet estimateur sur les compacts et nous calculons sa vitesse de convergence presque sûre.

Introduction

Extreme value analysis has attracted considerable attention in many fields of application, such as hydrology, biology and finance, for instance. The main result of extreme value theory asserts that the asymptotic distribution of the -properly rescaled -maximum of a sequence (Y 1 , . . . , Y n ) of independent copies of a random variable Y with distribution function F is a distribution having the form for some γ ∈ R, with G 0 (x) = exp(-e -x ). The distribution function F is then said to belong to the maximum domain of attraction of G γ and the parameter γ is called the extreme value index. Many applications in the areas of finance, insurance and geology, to name a few, can be found in the case when γ > 0, where F is a heavy-tailed distribution i.e. the associated survival function F := 1 -F satisfies F (x) = x -1/γ L(x), where γ shall now be referred to as the tail-index and L is a slowly varying function at infinity: namely, L satisfies, for all λ > 0, L(λx)/L(x) → 1 as x goes to infinity. In this case, the parameter γ clearly drives the tail behavior of F ; its estimation is in general a first step of extreme value analysis. For instance, if the idea is to estimate extreme quantiles -namely, quantiles with order α n > 1 -1/n, where n is the sample size -then one has to extrapolate beyond the available data using an extreme value model which depends on the tail-index. For this reason, the problem of estimating γ has been extensively studied in the literature. Recent overviews on univariate tail-index estimation can be found in the monographs of Beirlant et al. [START_REF] Beirlant | Statistics of extremes -Theory and applications[END_REF] and de Haan and Ferreira [START_REF] De Haan | Extreme value theory: An introduction[END_REF].

In practice, it is often useful to link the variable of interest Y to a covariate X. In this situation, the tail-index depends on the observed value x of the covariate X and shall be referred to, in the following, as the conditional tail-index. Its estimation has been addressed in the recent extreme value literature, albeit mostly when the covariates are nonrandom. Smith [START_REF] Smith | Extreme value analysis of environmental time series: an application to trend detection in ground-level ozone (with discussion)[END_REF] and Davison and Smith [START_REF] Davison | Models for exceedances over high thresholds[END_REF] considered a parametric regression model while Hall and Tajvidi [START_REF] Hall | Nonparametric analysis of temporal trend when fitting parametric models to extreme-value data[END_REF] used a semi-parametric approach to estimate the conditional tail-index. Fully nonparametric methods have been considered using splines (see Chavez-Demoulin and Davison [START_REF] Chavez-Demoulin | Generalized additive modelling of sample extremes[END_REF]), local polynomials (see Davison and Ramesh [9]), a moving window approach (see Gardes and Girard [START_REF] Gardes | A moving window approach for nonparametric estimation of the conditional tail index[END_REF]), or a nearest neighbor approach (see Gardes and Girard [START_REF] Gardes | Conditional extremes from heavy-tailed distributions: an application to the estimation of extreme rainfall return levels[END_REF]), among others.

Less attention though has been paid to the random covariate case, despite its practical interest. One can recall the works of Wang and Tsai [START_REF] Wang | Tail index regression[END_REF], based on a maximum likelihood approach in the Hall class of distribution functions (see Hall [START_REF] Hall | On some simple estimates of an exponent of regular variation[END_REF]), Daouia et al. [START_REF] Daouia | Kernel estimators of extreme level curves[END_REF] who use a fixed number of nonparametric conditional quantile estimators to estimate the conditional tail-index, later generalized in Daouia et al. [START_REF] Daouia | On kernel smoothing for extremal quantile regression[END_REF] to a regression context with response distributions belonging to the general max-domain of attraction, and Goegebeur et al. [START_REF] Goegebeur | Nonparametric regression estimation of conditional tails -the random covariate case, to appear in Statistics[END_REF] and Gardes and Stupfler [START_REF] Gardes | Estimation of the conditional tail index using a smoothed local Hill estimator[END_REF] who both provide adaptations of Hill's estimator (Hill [22]), the latter also studying an average of Hill-type statistics to improve the finite sample performance of the method.

In this paper, we focus on a nonparametric regression estimator of conditional tails introduced by Goegebeur et al. [START_REF] Goegebeur | Nonparametric regression estimation of conditional tails -the random covariate case, to appear in Statistics[END_REF]. The particular structure of this estimator makes it possible to study its uniform properties. Note that uniform properties of estimators of the conditional tail-index are seldom considered in the literature. One can think of the work of Gardes and Stupfler [START_REF] Gardes | Estimation of the conditional tail index using a smoothed local Hill estimator[END_REF], who study the uniform weak consistency of their estimator. Outside the field of conditional tail-index estimation, uniform convergence of the Parzen-Rosenblatt density estimator (Parzen [START_REF] Parzen | On estimation of a probability density function and mode[END_REF] and Rosenblatt [START_REF] Rosenblatt | Remarks on some nonparametric estimates of a density function[END_REF]) was first considered by Nadaraya [START_REF] Nadaraya | On non-parametric estimates of density functions and regression curves[END_REF]. His results were then improved by Silverman [START_REF] Silverman | Weak and strong uniform consistency of the kernel estimate of a density and its derivatives[END_REF] and Stute [START_REF] Stute | A law of the iterated logarithm for kernel density estimators[END_REF], the latter proving a law of the iterated logarithm in this context. Analogous results on kernel regression estimators were obtained by, among others, Mack and Silverman [START_REF] Mack | Weak and strong uniform consistency of kernel regression estimates[END_REF], Härdle et al. [START_REF] Härdle | Strong uniform consistency rates for estimators of conditional functionals[END_REF] and Einmahl and Mason [START_REF] Einmahl | An empirical process approach to the uniform consistency of kernel-type function estimators[END_REF]. Uniform consistency of isotonized versions of order-α quantile estimators introduced in Aragon et al. [START_REF] Aragon | Nonparametric frontier estimation: a conditional quantile-based approach[END_REF] was shown in Daouia and Simar [START_REF] Daouia | Robust nonparametric estimators of monotone boundaries[END_REF]. The case of estimators of the left-truncated quantiles is considered in Lemdani et al. [START_REF] Lemdani | Asymptotic properties of a conditional quantile estimator with randomly truncated data[END_REF]. Finally, the uniform strong consistency of a frontier estimator using kernel regression on high order moments was shown in Girard et al. [START_REF] Girard | Uniform strong consistency of a frontier estimator using kernel regression on high order moments[END_REF].

The paper is organised as follows. Our main results are stated in Section 2. The estimator is shown to be uniformly strongly consistent on compact sets in a semiparametric framework. The rate of convergence is provided when a further condition on the bias is satisfied. The rate of uniform convergence is closely linked to the rate of pointwise convergence in distribution established in Goegebeur et al. [START_REF] Goegebeur | Nonparametric regression estimation of conditional tails -the random covariate case, to appear in Statistics[END_REF]. The proofs of the main results are given in Section 3. Auxiliary results are postponed to the Appendix.

Main results

We assume that the covariate X takes its values in R d for some d ≥ 1. We shall work in the following semiparametric framework: (SP ) X has a probability density function f with support S ⊂ R d having nonempty interior and the conditional survival function of Y given X = x is such that

∀ x ∈ S, ∀ y ≥ 1, F (y | x) = y -1/γ(x) L(y | x)
where γ(x) > 0 and L(• | x) is a slowly varying function at infinity. The estimator of the conditional tail-index we shall study in this paper is defined as

γ n (x) := n i=1 K h (x -X i )(log Y i -log ω n,x ) + 1l {Yi>ωn,x} n i=1 K h (x -X i )1l {Yi>ωn,x}
.

(

Here K h (u) := h -d K(u/h) where K is a probability density function on R d and h := h n is a positive sequence tending to 0 while for all x, (ω n,x ) is a positive sequence tending to infinity. Note that γ n (x) = T

(1)

n (x)/T (0) n (x)
where, for all t ≥ 0,

T (t) n (x) := 1 n n i=1 K h (x -X i )(log Y i -log ω n,x ) t + 1l {Yi>ωn,x} .
The estimator ( 1) is an element of the family of estimators introduced in Goegebeur et al. [START_REF] Goegebeur | Nonparametric regression estimation of conditional tails -the random covariate case, to appear in Statistics[END_REF], which can be seen as an adaptation of the classical Hill estimator of the tail-index for univariate distributions (see Hill [START_REF] Hill | A simple general approach to inference about the tail of a distribution[END_REF]). Note that the threshold ω n,x is local, i.e. it depends on the point x where the estimation is to be made, while the bandwidth h is global.

We first wish to state the uniform strong consistency of our estimator on an arbitrary compact subset Ω of R d contained in the interior of S. To this end, we first assume that for every x ∈ S the slowly varying function

L(• | x) appearing in F (• | x) is normalised (see Bingham et al. [3]):
(A 1 ) For all x ∈ S and y ≥ 1,

L(y | x) = c L (x) exp y 1 α(v | x) v dv
where c L (x) > 0 and α(• | x) is a function converging to 0 at infinity.

Let • be a norm on R d and for r > 0, let Ω r be the set of those points in R d whose distance to Ω is not more than r:

Ω r = {x ∈ R d | ∃x ′ ∈ Ω, x -x ′ ≤ r}.
Remark that since Ω is contained in the interior of the closed set S, the distance of Ω to the boundary of S must be positive. As a consequence, the set Ω r is contained in S for all r > 0 small enough. We can therefore introduce some classical regularity assumptions:

(A 2 ) For some r > 0, on Ω r , the functions f and γ are positive Hölder continuous functions, log c L is a Hölder continuous function and α(y

| •) is a Hölder continuous function uniformly in y ≥ 1: for all x, x ′ ∈ Ω r , |f (x) -f (x ′ )| ≤ M f x -x ′ η f , |γ(x) -γ(x ′ )| ≤ M γ x -x ′ ηγ , | log c L (x) -log c L (x ′ )| ≤ M cL x -x ′ ηc L , sup y≥1 |α(y | x) -α(y | x ′ )| ≤ M α x -x ′ ηα .
Let moreover η := η γ ∧ η cL ∧ η α . We introduce the oscillation of x → log ω n,x at a point x ∈ R d over the ball B(x, ε):

∀ ε > 0, ∆(log ω n,x )(ε) := sup z∈B(x, ε) |log ω n,x -log ω n,z | imsart-
x, x ′ ∈ R d , |K(x) -K(x ′ )| ≤ M K x -x ′ ηK
and its support is included in the unit ball B of R d .

Especially, if (K) holds then K is bounded with compact support. Let

v n (x) := nh d log n F (ω n,x | x)
and introduce the hypothesis (C) For some b > 0, it holds that lim sup

n→∞ sup x∈Ω v n (x)∆(log ω n,x )(n -b ) < ∞.
Our uniform strong consistency result may now be stated:

Theorem 1. Assume that (SP ), (K), (A 1 ) and (A 2 ) hold and that

• inf x∈Ω v n (x) → ∞; • inf x∈Ω ω n,x → ∞; • h η sup x∈Ω log ω n,x → 0; • sup x∈Ω ∆(log ω n,x )(h) → 0; • sup x∈Ω α(y | x) → 0 as y → ∞.
ω n,x ω n,z -1 ≤ 2 sup x∈Ω ∆(log ω n,x )(h) → 0. (2)
Finally, the conditions x) where g : S → R is a positive Hölder continuous function whose Hölder exponent is not less than η. In other words, Theorem 1 requires that a continuity property on x → log ω n,x be satisfied.

sup x∈Ω ∆(log ω n,x )(h) → 0 and lim sup n→∞ sup x∈Ω v n (x)∆(log ω n,x )(n -b ) < ∞ are satisfied if for instance ω n,x = n g(
Our second aim is to compute the rate of uniform strong consistency of the estimator (1):

v n (x) {α(ω n,x | x) ∨ h η f ∨ h η log ω n,x ∨ ∆(log ω n,x )(h)} < ∞ (3)
then it holds that sup

x∈Ω v n (x) | γ n (x) -γ(x)| = O (1) almost surely as n → ∞.
Let us highlight that condition (3) controls the bias of the estimator γ n . The terms h η f and h η log ω n,x correspond to the bias which stems from the use of a kernel regression, while the presence of the other terms is due to the particular structure of the semiparametric model (SP ). Besides, as pointed out in Goegebeur et al. [START_REF] Goegebeur | Nonparametric regression estimation of conditional tails -the random covariate case, to appear in Statistics[END_REF], the rate of pointwise convergence of

γ n (x) to γ(x) is [nh d F (ω n,x | x)] 1/2 . Up to the term [log n] 1/2
, the rate of uniform convergence of γ n to γ is therefore the infimum (over Ω) of the rate of pointwise convergence of γ n (x) to γ(x).

Proofs of the main results

Before starting the proof of Theorem 1, let us note that assuming that (SP ), (A 1 ) and (A 2 ) hold then it is easy to show that there exists a positive constant M F such that the function (x, y) → log F (y | x) has the following property: for all x, x ′ ∈ Ω r such that x -x ′ ≤ 1 and y, y

′ ≥ e, log F (y | x) F (y ′ | x ′ ) ≤ M F x -x ′ η log y + 1 γ(x ′ ) + α(y ∧ y ′ | x ′ ) | log y -log y ′ |. (4)
Moreover, if (A 2 ) holds then one may take a positive number r such that the four conditions of the hypothesis hold on Ω 2r . Since Ω r is compact, f := sup Ω r f < ∞ and f := inf Ω r f > 0. As a consequence, the uniform relative oscillation of f over the ball B(x, h) can be controlled as

sup x∈Ω r sup z∈B(x, h) f (z) f (x) -1 = O (h η f ) → 0. (5)
Second, γ := sup Ω r γ < ∞ and γ := inf Ω r γ > 0 and we thus have

sup x∈Ω r sup z∈B(x, h) γ(z) γ(x) -1 = O (h ηγ ) → 0. (6)
Third, we can write for all x, x ′ ∈ Ω r and t ≥ 1

α(t | x) ≤ α(t | x ′ ) + |α(t | x) -α(t | x ′ )|
and the roles of x and x ′ are symmetric in the above inequality, so that taking the supremum over t ≥ y on both sides yields

∀ y ≥ 1, |α(y | x) -α(y | x ′ )| ≤ M α x -x ′ ηα . (7)
We may now prove the key result for the proof of Theorem 1, which is a uniform law of large numbers for

T (0) n (x) and T (1) n (x).
In what follows, we let µ

(t) n (x) := E(T (t) n (x)).
Proposition 1. Assume that the conditions of Theorem 1 are satisfied. Then for every t ∈ {0, 1} it holds that Proof of Proposition 1. The proof is based on that of Lemma 1 in Härdle and Marron [START_REF] Härdle | Optimal bandwidth selection in nonparametric regression function estimation[END_REF]: we shall in fact show complete convergence in the sense of Hsu and Robbins [START_REF] Hsu | Complete convergence and the law of large numbers[END_REF]. Since Ω is a compact subset of R d , we may, for every n ∈ N \ {0}, find a finite subset Ω n of Ω such that:

sup x∈Ω v n (x) T (t) n (x) µ (t) n (x) -1 = O(1)
∀ x ∈ Ω, ∃ χ(x) ∈ Ω n , x -χ(x) ≤ n -b and ∃ c > 0, |Ω n | = O (n c ) ,
where b, which we may take to be not less than 1/d + 1/2η K , is given by condition (C) and |Ω n | stands for the cardinality of Ω n . Notice that, since nh d → ∞, one has n -b /h → 0, so that one can assume that eventually

χ(x) ∈ B(x, h) for all x ∈ Ω. Next, remark that x -χ(x) ≤ n -b ≤ h ≤ 1 and that since n -b ≤ h the convergences n -bη sup x∈Ω log ω n,x ≤ h η sup x∈Ω log ω n,x → 0 and sup x∈Ω ∆(log ω n,x )(n -b ) ≤ sup x∈Ω ∆(log ω n,x )(h) → 0 hold. Consequently, Lemma 1 entails sup x∈Ω v n (x) v n (χ(x)) -1 = sup x∈Ω F (ω n,x | x) F (ω n,χ(x) | χ(x)) -1 → 0. ( 8
)
Pick ε > 0 and an arbitrary sequence of positive numbers (δ n ) converging to 0; using together ( 8) and the triangular inequality thus shows that for n large enough

P δ n sup x∈Ω v n (x) T (t) n (x) µ (t) n (x) -1 > ε ≤ R 1,n + R 2,n
where

R 1,n := z∈Ωn P δ n v n (z) T (t) n (z) µ (t) n (z) -1 > ε 4 and R 2,n := P δ n sup x∈Ω v n (x) T (t) n (x) µ (t) n (x) - T (t) n (χ(x)) µ (t) n (χ(x)) > ε 2 .
The goal of the proof is now to show that the series n R 1,n and n R 2,n converge. The result of Proposition 1 shall then be an easy consequence of Borel-Cantelli's lemma and Lemma 6.

We start by controlling R 1,n . To this end, apply Lemma 3 to get that there exists a positive constant κ such that for n large enough,

∀ z ∈ Ω n , P δ n v n (z) T (t) n (z) µ (t) n (z) -1 > ε 4 ≤ 2 exp - κ 16 ε 2 nh d F (ω n,z | z) δ 2 n v 2 n (z)
.

Use now the definition of v n (z) to get

R 1,n = O n c exp - κ 16 ε 2 log n δ 2 n .
Hence n R 1,n converges.

We now turn to R 2,n . Using the triangular inequality gives

R 2,n ≤ P δ n sup x∈Ω v n (x)S 1,n (x) > ε 4 + P δ n sup x∈Ω v n (x)S 2,n (x) > ε 4 =: R 3,n + R 4,n
where and it is enough to show that the series n R 3,n and n R 4,n converge.

S 1,n (x) := 1 n n i=1 K h (x -X i ) µ (t) n (x) - K h (χ(x) -X i ) µ (t) n (χ(x)) (log Y i -log ω n,χ(x) ) t + 1l {Yi>ω n,χ(x) } , S 2,n (x) := 1 n n i=1 K h (x -X i ) µ (t) n (x) (log Y i -log ω n,x ) t + 1l {Yi>ωn,x} -(log Y i -log ω n,χ(x) ) t + 1l {Yi>ω n,χ(x) } , imsart-
To deal with n R 3, n use once again the triangular inequality to obtain

µ (t) n (χ(x)) K h (x -X i ) µ (t) n (x) - K h (χ(x) -X i ) µ (t) n (χ(x)) ≤ |K h (x -X i ) -K h (χ(x) -X i )| + µ (t) n (χ(x)) µ (t) n (x) -1 K h (x -X i ).
Using hypothesis (K) and Lemma 4, there exists a positive constant M such that for n large enough:

∀ x ∈ Ω, µ (t) n (χ(x)) K h (x -X i ) µ (t) n (x) - K h (χ(x) -X i ) µ (t) n (χ(x)) ≤ M h d n -b h ηK ∨ ∆(log ω n,x )(n -b ) .
Besides

m (t) n (z) := 1 n n i=1 K 2h (z -X i )(log Y i -log ω n,z ) t + 1l {Yi>ωn,z}
is the empirical analogue of m

(t)
n (z) defined before Lemma 4; since the support of the random variable

K h (x -X i ) is included in B(χ(x), 2h), one has for n large enough ∀ x ∈ Ω, v n (x)S 1,n (x) ≤ 2 d VM v n (x) n -b h ηK ∨ ∆(log ω n,x )(n -b ) m (t) n (χ(x)) µ (t) n (χ(x)) . Moreover, since m (t) n (z) is a kernel estimator of m (t)
n (z, z) for which the conditions of Lemma 2 are satisfied, we get for n large enough:

∀ z ∈ Ω n , δ n v n (z) m (t) n (z) µ (t) n (z) ≤ 2δ n v n (z) 1 + m (t) n (z) m (t) n (z) -1 . The fact that b ≥ 1/d + 1/2η K gives sup z∈Ωn v n (z) n -b h ηK ≤ √ n n -b h ηK ≤ 1 nh d ηK /d → 0.
Using first this convergence together with hypothesis (C) and ( 8) and then Lemma 3 entails for n large enough:

R 3,n ≤ z∈Ωn P δ n v n (z) m (t) n (z) m (t) n (z) -1 > ε = O n c exp -κ ′ ε 2 log n δ 2 n
where κ ′ is a positive constant. Hence n R 3,n converges.

To control n R 4,n first use Lemmas 2(iv) and 4 to get, for n large enough

sup x∈Ω m (t) n (χ(x)) µ (t) n (x) = sup x∈Ω m (t) n (χ(x)) µ (t) n (χ(x)) µ (t) n (χ(x)) µ (t) n (x) ≤ 2.
Therefore, since the support of the random variable K h (x -X i ) is included in B(χ(x), 2h), one has for n large enough and all

x ∈ Ω S 2,n (x) ≤ 2 d+1 V K ∞ S 3,n (x)
where

K ∞ := sup B K and
S 3,n (x) := 1 n n i=1 K 2h (χ(x) -X i ) m (t) n (χ(x)) (log Y i -log ω n,x ) t + 1l {Yi>ωn,x} -(log Y i -log ω n,χ(x) ) t + 1l {Yi>ω n,χ(x) } .
imsart-aihp ver. 2013/03/06 file: Hill_loiunif_IHP_revised6.tex date: November 21, 2013

We then get

R 4,n ≤ P δ n sup x∈Ω v n (x)S 3,n (x) > ε 2 d+3 V K ∞ =: R 5,n
and it is enough to control n R 5,n . We start by considering the case t = 0. In this case, S 3,n (x) reduces to

S 3,n (x) = 1 n n i=1 K 2h (χ(x) -X i ) m (0) n (χ(x)) 1l {ωn,x∧ω n,χ(x) <Yi≤ωn,x∨ω n,χ(x) } .
Letting ρ n,x := 2∆(log ω n,x )(n -b ) and using (2), we have sup x∈Ω ρ n,x → 0 and for n large enough

∀ x ∈ Ω, (1 -ρ n,χ(x) )ω n,χ(x) ≤ ω n,x ≤ (1 + ρ n,χ(x) )ω n,χ(x) .
As a consequence, for n large enough it holds that

∀ x ∈ Ω, S 3,n (x) ≤ 1 n n i=1 K 2h (χ(x) -X i ) m (0) n (χ(x)) 1l {(1-ρ n,χ(x) )ω n,χ(x) <Yi≤(1+ρ n,χ(x) )ω n,χ(x) } .
Similarly to Lemma 5, let

M n (x) := E(K 2h (x -X)1l {(1-ρn,x)ωn,x<Y <(1+ρn,x)ωn,x} ) and U n (x) := 1 n n i=1 K 2h (x -X i )1l {(1-ρn,x)ωn,x<Yi<(1+ρn,x)ωn,x} . Write ∀ x ∈ Ω, δ n v n (x)S 3,n (x) ≤ δ n v n (x) M n (χ(x)) m (0) n (χ(x)) 1 + U n (χ(x)) M n (χ(x)) -1 .
Use together Lemmas 2(iv) and 5 along with [START_REF] Daouia | Robust nonparametric estimators of monotone boundaries[END_REF] to get for n large enough

∀ x ∈ Ω, δ n v n (x)S 3,n (x) ≤ 4 γ(χ(x)) δ n v n (χ(x))ρ n,χ(x) 1 + U n (χ(x)) M n (χ(x)) -1 .
Recall that ρ n,x = 2∆(log ω n,x )(n -b ) and that condition (C) is satisfied to obtain

δ n sup z∈Ωn v n (z)ρ n,z → 0.
Therefore, since 0 < γ ≤ γ(χ(x)), the triangular inequality implies that

R 5,n ≤ z∈Ωn P δ n v n (z)ρ n,z U n (z) M n (z) -1 > εγ 2 d+6 V K ∞ for n large enough. Lemma 5 now makes it clear that R 5,n = O n c sup z∈Ωn exp -κ ′′ εγ 2 d+6 V K ∞ v n (z) log n δ n = o n c exp -κ ′′ ε log n δ n
which proves that n R 5,n converges in this case.

If now t = 1, we recall (45) in the proof of Lemma 4 to get for n large enough and for all x ∈ Ω

S 3,n (x) = log ω n,x ω n,χ(x) m (0) n (χ(x)) m (1) n (χ(x)) 1 n n i=1 K 2h (χ(x) -X i ) m (0) n (χ(x))
1l Use (2) and Lemma 2(iv) to get for n large enough

∀ x ∈ Ω, S 3,n (x) ≤ 2 γ ∆(log ω n,x )(n -b ) 1 n n i=1 K 2h (χ(x) -X i ) m (0) n (χ(x)) 1l {Yi>ω n,χ(x) /2} ≤ 2 γ ∆(log ω n,x )(n -b ) ν n (χ(x)) m (0) n (χ(x)) 1 + V n (χ(x)) ν n (χ(x)) -1 (9)
where

ν n (x) := E(K 2h (x -X)1l {Y >ωn,x/2} ) and V n (x) := 1 n n i=1 K 2h (x -X i )1l {Yi>ωn,x/2} .
The family of sequences (ω n,x /2) clearly satisfies the hypotheses of Lemmas 2 and 3: in particular

sup x∈Ω ν n (x) m (0) n (x) F (ω n,x | x) F (ω n,x /2 | x) -1 → 0 ( 10
)
and there exists a positive constant κ ′′′ such that for n large enough

∀ x ∈ Ω, P V n (x) ν n (x) -1 > ε ≤ 2 exp -κ ′′′ ε 2 nh d F (ω n,x | x) (11)
where the inequality F (ω n,x /2 | x) ≥ F (ω n,x | x) was used. We conclude by noting that according to (4), lim sup

n→∞ sup x∈Ω log F (ω n,x | x) F (ω n,x /2 | x) ≤ log 2 γ < ∞ ⇒ 0 < lim sup n→∞ sup x∈Ω F (ω n,x | x) F (ω n,x /2 | x) < ∞.
This property together with (10) entails the convergences

δ n sup x∈Ω ν n (x) m (0) n (x) → 0 and sup x∈Ω ∆(log ω n,x )(n -b ) ν n (x) m (0) n (x) → 0. ( 12
)
Reporting [START_REF] Davison | Models for exceedances over high thresholds[END_REF] along with ( 12) into (9), recalling condition (C) and using the triangular inequality together with [START_REF] Daouia | Robust nonparametric estimators of monotone boundaries[END_REF] shows that for n large enough,

R 5,n ≤ z∈Ωn P δ n v n (z) V n (x) ν n (x) -1 > ε = O n c exp -κ ′′′ ε 2 log n δ 2 n
where [START_REF] Einmahl | An empirical process approach to the uniform consistency of kernel-type function estimators[END_REF] was used in the last step. As a consequence, n R 5,n converges in this case as well. This completes the proof of Proposition 1.

With Proposition 1 at hand, we can now prove Theorem 1 and Theorem 2.

Proof of Theorem 1. Notice that

γ n (x) = µ (1) n (x) µ (0) n (x) T (1) n (x) µ (1) n (x) µ (0) n (x) T (0) n (x) . ( 13
)
Applying Proposition 1 twice yields

sup x∈Ω T (1)
n (x) µ

(1)

n (x) µ (0) n (x) T (0) n (x) -1 → 0 almost surely as n → ∞. (14)
Moreover, recalling that γ is continuous and therefore bounded on the compact set Ω, using Lemma 2(i) and (iv) twice entails

sup x∈Ω µ (1) n (x) µ (0) n (x) -γ(x) → 0 as n → ∞. (15)
The result follows by reporting ( 14) and ( 15) into [START_REF] Gardes | Conditional extremes from heavy-tailed distributions: an application to the estimation of extreme rainfall return levels[END_REF].

n (x) µ

(1)

n (x) -1 = O(1) (16)
almost surely as n → ∞. Moreover, Lemma 2 (iv) gives

sup x∈Ω 1 α(ω n,x | x) ∨ h η f ∨ h η log ω n,x µ (t) n (x) f (x)F (ω n,x | x) -γ t (x) = O(1)
for t ∈ {0, 1}, so that using condition (3),

sup x∈Ω v n (x) µ (1) n (x) µ (0) n (x) -γ(x) = O(1). ( 17
)
The result follows by reporting ( 16) and ( 17) into [START_REF] Gardes | Conditional extremes from heavy-tailed distributions: an application to the estimation of extreme rainfall return levels[END_REF].

Appendix: Auxiliary results and proofs

The first lemma of this section is a technical result that gives an upper bound for the oscillation of the log-conditional survival function.

Lemma 1. Assume that (SP ), (A 1 ) and (A 2 ) hold. Let moreover ε := ε n , ε ′ := ε ′ n and ε ′′ := ε ′′ n be three positive sequences tending to 0 and assume that

• inf x∈Ω ω n,x → ∞ ; • ε ′′η sup x∈Ω log ω n,x → 0 ; • sup x∈Ω ∆(log ω n,x )(ε ′ ) → 0 ; • sup x∈Ω α(y | x) → 0 as y → ∞.
Then it holds that, for n large enough,

∀ (x, x ′ ) ∈ Ω × Ω ε , ∀(z, z ′ ) ∈ B(x, ε ′ ) × B(x ′ , ε ′′ ), log F (ω n,z | z ′ ) F (ω n,x | x ′ ) ≤ M F ε ′′η log ω n,z + 2 γ ∆(log ω n,x )(ε ′ ).
In particular, For every y ≥ 1, inequality (7) entails sup

sup x∈Ω sup x ′ ∈Ω ε sup z∈B(x, ε ′ ) sup z ′ ∈B(x ′ , ε ′′ ) 1 ε ′′η log ω n,x ∨ ∆(log ω n,x )(ε ′ ) F (ω n,z | z ′ ) F (ω n,x | x ′ ) -1 = O(1). Proof of Lemma 1. Pick (x, x ′ ) ∈ Ω × Ω ε and (z, z ′ ) ∈ B(x, ε ′ ) × B(x ′ , ε ′′ ). Use (4) to get for n large enough log F (ω n,z | z ′ ) F (ω n,x | x ′ ) ≤ M F x ′ -z ′ η log ω n,z + 1 γ(x ′ ) + α(ω n,z ∧ ω n,x | x ′ ) |log ω n,x -log ω n,z | .
x∈Ω ε α(y | x) ≤ sup x∈Ω α(y | x) + M α ε ηα . ( 18
)
Using then (2) with ε ′ instead of h, we get inf

x∈Ω inf z∈B(x, ε ′ ) ω n,z ∧ ω n,x = inf x∈Ω ω n,x (1 + o(1)) → ∞, so that sup x∈Ω sup x ′ ∈Ω ε sup z∈B(x, ε ′ ) α(ω n,z ∧ ω n,x | x ′ ) → 0.
Especially, since 0 < γ ≤ γ(x ′ ), we obtain for n large enough:

∀ (x, x ′ ) ∈ Ω × Ω ε , ∀(z, z ′ ) ∈ B(x, ε ′ ) × B(x ′ , ε ′′ ), log F (ω n,z | z ′ ) F (ω n,x | x ′ ) ≤ M F ε ′′η log ω n,z + 2 γ ∆(log ω n,x )(ε ′ )
which is the first part of the result. To prove the second part, note that because sup

x∈Ω ∆(log ω n,x )(ε ′ ) → 0 it
holds that for n large enough

∀ (x, x ′ ) ∈ Ω × Ω ε , ∀(z, z ′ ) ∈ B(x, ε ′ ) × B(x ′ , ε ′′ ), log F (ω n,z | z ′ ) F (ω n,x | x ′ ) ≤ 2M F ε ′′η log ω n,x + 2 γ ∆(log ω n,x )(ε ′ ). Consequently sup x∈Ω sup x ′ ∈Ω ε sup z∈B(x, ε ′ ) sup z ′ ∈B(x ′ , ε ′′ ) 1 ε ′′η log ω n,x ∨ ∆(log ω n,x )(ε ′ ) log F (ω n,z | z ′ ) F (ω n,x | x ′ ) = O(1).
Using the equivalent e u -1 = u(1 + o(1)) therefore completes the proof of Lemma 1.

The second lemma examines the behavior of the conditional moment

m (t) n (x, z) := E((log Y -log ω n,x ) t + 1l {Y >ωn,x} | X = z)
and that of its smoothed version µ

(t) n (x) = E(K h (x -X)m (t)
n (x, X)). Let Γ be Euler's Gamma function:

∀ t > 0, Γ(t) := +∞ 0 v t-1 e -v dv.
Lemma 2. Assume that (SP ), (A 1 ) and (A 2 ) hold. Pick t ≥ 0 and assume that K is a bounded probability density function on R d with support included in B. If moreover

• inf x∈Ω ω n,x → ∞ ; • h η sup x∈Ω log ω n,x → 0 ; • sup x∈Ω α(y | x) → 0 as y → ∞
then, as n → ∞, the following estimations hold:

(i) sup x∈Ω sup z∈B(x, h) 1 α(ω n,x | x) ∨ h ηα m (t) n (x, z) γ t (z)Γ(t + 1)F (ω n,x | z) -1 = O (1). (ii) sup x∈Ω sup z∈B(x,h) 1 α(ω n,x | x) ∨ h η log ω n,x m (t) n (x, z) m (t) n (x, x) -1 = O (1). (iii) sup x∈Ω 1 α(ω n,x | x) ∨ h η f ∨ h η log ω n,x µ (t) n (x) f (x)m (t) n (x, x) -1 = O (1
(iv) sup x∈Ω 1 α(ω n,x | x) ∨ h η f ∨ h η log ω n,x µ (t) n (x) f (x)γ t (x)Γ(t + 1)F (ω n,x | x) -1 = O (1).
Proof of Lemma 2. (i) When t = 0, there is nothing to prove, since m (0) n (x, z) = F (ω n,x | z) and Γ(1) = 1. In the case t > 0, an integration by parts yields

m (t) n (x, z) = +∞ ωn,x t (log y -log ω n,x ) t-1 y F (y | z) dy = t F (ω n,x | z) +∞ 1 (log r) t-1 F (rω n,x | z) rF (ω n,x | z) dr.
From (SP ) and (A 1 ), one has

F (rω n,x | z) rF (ω n,x | z) -r -1/γ(z)-1 = r -1/γ(z)-1 exp rωn,x ωn,x α(v | z) v dv -1 . (19)
For all y ∈ R, the mean value theorem yields |e y -1| ≤ |y|e |y| . Meanwhile,

rωn,x ωn,x α(v | z) v dv ≤ α(ω n,x | z) log r. (20)
Choosing n so large that sup 18), ( 19) and ( 20) together imply that, for all

x∈Ω sup z∈B(x, h) α(ω n,x | z) < 1/2γ, (
x ∈ Ω and z ∈ B(x, h),

+∞ 1 (log r) t-1 F (rω n,x | z) rF (ω n,x | z) -r -1/γ(z)-1 dr ≤ (α(ω n,x | x) + M α h ηα ) +∞ 1 (log r) t r -1/2γ-1 dr
which, since the integral on the right-hand side of this inequality converges, gives

sup x∈Ω sup z∈B(x, h) 1 α(ω n,x | x) ∨ h ηα +∞ 1 (log r) t-1 F (rω n,x | z) rF (ω n,x | z) -r -1/γ(z)-1 dr = O (1)
as n → ∞. An elementary change of variables and the well-known equality tΓ(t) = Γ(t + 1) thus entail

sup x∈Ω sup z∈B(x, h) 1 α(ω n,x | x) ∨ h ηα m (t) n (x, z) F (ω n,x | z) -γ t (z)Γ(t + 1) = O (1)
as n → ∞ and (i) is proven.

(ii) Since for all x ∈ Ω, 0

< γ ≤ γ(x) ≤ γ < ∞, applying (i) entails sup x∈Ω sup z∈B(x,h) 1 α(ω n,x | x) ∨ h ηα m (t) n (x, z) γ t (z)Γ(t + 1)F (ω n,x | z) γ t (x)Γ(t + 1)F (ω n,x | x) m (t) n (x, x) -1 = O (1) . (21)
Moreover, hypothesis (A 2 ) and the mean value theorem yield

γ t (x) γ t (z) -1 ≤ 1 γ t sup γ≤r≤γ tr t-1 sup x∈Ω sup z∈B(x,h) |γ(x) -γ(z)| = O(h ηγ ). (22)
Besides, using Lemma 1 gives

sup x∈Ω sup z∈B(x,h) 1 h η log ω n,x F (ω n,x | x) F (ω n,x | z) -1 = O(1). ( 23
)
imsart-aihp ver. 2013/03/06 file: Hill_loiunif_IHP_revised6.tex date: November 21, 2013

Note finally that since η ≤ η γ ∧ η α and inf

x∈Ω ω n,x → ∞ one has sup x∈Ω h ηγ ∨ h ηα h η log ω n,x → 0.
Using then ( 22) and ( 23) together with ( 21) yields (ii).

(iii) Let us remark that for all x ∈ Ω:

µ (t) n (x) f (x) m (t) n (x, x) = B K(u) f (x -hu) f (x) m (t) n (x, x -hu) m (t) n (x, x) du.
From ( 5) and (ii) it follows that

sup x∈Ω sup z∈B(x, h) 1 α(ω n,x | x) ∨ h η f ∨ h η log ω n,x f (z) f (x) m (t) n (x, z) m (t) n (x, x) -1 → 0 as n → ∞, which yields (iii).
(iv) This is a straightforward consequence of (i) and (iii).

The third lemma is essential to prove Proposition 1. It gives a uniform exponential bound for large deviations of T

n and T

n .

Lemma 3. Assume that (SP ), (A 1 ) and (A 2 ) hold. Assume that K is a bounded probability density function on R d with support included in B. If moreover

• inf x∈Ω ω n,x → ∞ ; • h η sup x∈Ω log ω n,x → 0 ; • sup x∈Ω α(y | x) → 0 as y → ∞
then there exists a positive constant κ such that for all n large enough, one has for t ∈ {0, 1} and every ε > 0 small enough:

∀ x ∈ Ω, P T (t) n (x) µ (t) n (x) -1 > ε ≤ 2 exp -κε 2 nh d F (ω n,x | x) .
Proof of Lemma 3. For every x ∈ Ω:

P T (0) n (x) µ (0) n (x) -1 > ε = P h d T (0) n (x) -h d µ (0) n (x) > εh d µ (0) n (x) . Notice now that if W n,i (x) := h d K h (x -X i )1l {Yi>ωn,x} then h d T (0) n (x) -h d µ (0) n (x) = 1 n n i=1 [W n,i (x) -E(W n,i (x))]
is a mean of bounded, centered, independent and identically distributed random variables. Define

τ n (x) := ε K ∞ nh d µ (0) n (x) and λ n (x) := ε K ∞ h d µ (0) n (x) Var(W n, 1 (x)) .
Bernstein's inequality (see Hoeffding [START_REF] Hoeffding | Probability inequalities for sums of bounded random variables[END_REF]) yields, for all ε > 0:

P T (0) n (x) µ (0) n (x) -1 > ε ≤ 2 exp - τ n (x)λ n (x) 2(1 + λ n (x)/3)
.

Applying Lemma 2(iii) yields for n large enough:

inf x∈Ω τ n (x) nh d F (ω n,x | x) ≥ εf 2 K ∞ . ( 24
)
Moreover, since W n, 1 (x) is bounded by K ∞ , it follows from the inequality

W 2 n, 1 (x) ≤ K ∞ W n, 1 (x) that sup x∈Ω 1 λ n (x) ≤ sup x∈Ω E(W 2 n, 1 (x)) ε K ∞ h d µ (0) n (x) ≤ 1 ε . ( 25
)
Finally, it holds that

τ n (x)λ n (x) 2(1 + λ n (x)/3) ≥ inf x∈Ω τ n (x) nh d F (ω n,x | x) inf x∈Ω 1 2(1/λ n (x) + 1/3) nh d F (ω n,x | x).
Using ( 24), ( 25) and the fact that the function t → 1/[2(t + 1/3)] is decreasing on R + , it is then clear that for all n large enough, if ε > 0 is small enough, there exists a positive constant κ 1 that is independent of ε such that

∀ x ∈ Ω, P T (0) n (x) µ (0) n (x) -1 > ε ≤ 2 exp -κ 1 ε 2 nh d F (ω n,x | x) .
We now turn to T

n (x). For every x ∈ Ω, it holds that P T

(1)

n (x) µ (1) n (x) -1 > ε = P T (1) n (x) µ (1) n (x) -1 > ε + P T (1) n (x) µ (1) n (x) -1 < -ε =: u 1,n (x) + u 2,n (x).
We shall then give a uniform Chernoff-type exponential bound (see Chernoff [START_REF] Chernoff | A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations[END_REF]) for both terms on the right-hand side of the above inequality. We start by considering u 1,n (x). Let

ϕ n (s, x) := E(exp(sK h (x -X)(log Y -log ω n,x ) + 1l {Y >ωn,x} ))
be the moment generating function of the random variable K h (x-X)(log Y -log ω n,x ) + 1l {Y >ωn,x} . Markov's inequality entails, for every q > 0,

u 1,n (x) = P exp q T (1) n (x) µ (1) n (x) > exp(q[ε + 1]) ≤ exp -q[ε + 1] + n log ϕ n q nµ (1) n (x) , x . ( 26
)
Our goal is now to use inequality [START_REF] Mack | Weak and strong uniform consistency of kernel regression estimates[END_REF] with a suitable value q * (ε, x) for q. To this end, notice that

ϕ n (s, x) = R d \B(x, h) f (z) dz + B(x, h) ψ n (sK h (x -z) | x, z) f (z) dz where ψ n (s | x, z) := E(exp(s(log Y -log ω n,x ) + 1l {Y >ωn,x} ) | X = z)
is the conditional moment generating function of the random variable (log Y -log ω n,x ) + 1l {Y >ωn,x} given X = z. In particular, since f is a probability density function on R d ,

ϕ n (s, x) = 1 + B(x, h) [ψ n (sK h (x -z) | x, z) -1] f (z) dz. ( 27
)
imsart-aihp ver. 2013/03/06 file: Hill_loiunif_IHP_revised6.tex date: November 21, 2013

This equality makes it clear that it is enough to study the behavior of ψ n (• | x, z). One has

ψ n (s | x, z) = 1 -F (ω n,x | z) + E Y ω n,x s 1l {Y >ωn,x} | X = z .
From this we deduce that

ψ n (s | x, z) = 1 + F (ω n,x | z) +∞ 1 st s F (tω n,x | z) tF (ω n,x | z) dt.
A use of ( 19) and ( 20) therefore entails, for all s < 1/γ,

ψ n (s | x, z) = 1 + sF (ω n,x | z) 1 γ(z) -s -1 + R n (s | x, z) (28)
where

R n (s | x, z) satisfies, for all δ > 0, if n is large enough, sup x∈Ω sup z∈B(x, h) |R n (s | x, z)| ≤ sup x∈Ω sup z∈B(x, h) α(ω n,x | z) +∞ 1 v s-1/γ-1+δ log v dv.
Since by [START_REF] Hall | On some simple estimates of an exponent of regular variation[END_REF] it holds that sup

x∈Ω sup z∈B(x, h) α(ω n,x | z) → 0 we get, for all δ > 0: sup s<1/γ-δ sup x∈Ω sup z∈B(x, h) |R n (s | x, z)| → 0 (29)
as n → ∞. We shall now derive a suitable value for the parameter q. Given X = x, if the remainder term R n were identically 0, then one would have m

n (x, x) = γ(x)F (ω n,x | x) (1)
and thus an optimal value of q would be obtained by minimizing the function

q → -q[1 + ε] + n log 1 + q n 1 - q nF (ω n,x | x) -1 .
Straightforward but cumbersome computations lead to the optimal value

q ⋆ c,+ (ε) := nF (ω n,x | x) 2 -F (ω n,x | x) - 2 -F (ω n,x | x) 2 - 4ε ε + 1 1 -F (ω n,x | x) 2 1 -F (ω n,x | x) . (30)
Since we are mostly interested in what happens in the limit n → ∞ and ε → 0, we may examine the behavior of q ⋆ c,+ (ε) in this case. Using [START_REF] Silverman | Weak and strong uniform consistency of the kernel estimate of a density and its derivatives[END_REF], we get the following asymptotic equivalent

q * c,+ (ε) = nF (ω n,x | x) ε 2(ε + 1) . Note that since q * c,+ (ε)/[nm (1)
n (x, x)] = ε/[2γ(x)(ε + 1)
] is positive and converges to 0 as ε → 0, the moment generating function

ψ n (• | x, x) at q * c,+ (ε)/[nm (1)
n (x, x)] is well-defined and finite for ε small enough and therefore this choice of q is valid. Back to our original context, taking into account the presence of the covariate X motivates the following value for q:

q * n,+ (ε, x) := M ε ε + 1 nh d f (x)F (ω n,x | x)
where M is a positive constant to be chosen later. For ε small enough and for n so large that the quantity

ϕ n q * n,+ (ε, x)/(nµ (1)
n (x)), x is well-defined and finite for all x ∈ Ω, replacing q by q * n,+ (ε, x) in the righthand side of [START_REF] Mack | Weak and strong uniform consistency of kernel regression estimates[END_REF] gives

∀ x ∈ Ω, u 1,n (x) ≤ exp -M εnh d f (x)F (ω n,x | x) + n log ϕ n q * n,+ (ε, x) nµ (1) n (x) , x . (31)
Using the classical inequality log(1 + r) ≤ r for all r > 0 together with ( 27) and ( 28), we obtain

log ϕ n (s, x) ≤ B(x, h) [ψ n (sK h (x -z) | x, z) -1] f (z) dz ≤ B(x, h) sK h (x -z)F (ω n,x | z) 1 γ(z) -sK h (x -z) -1 + R n (sK h (x -z) | x, z) f (z) dz.
According to Lemma 2(iv),

q * n,+ (ε, x) nµ (1) n (x)
= M ε ε + 1 h d f (x) F (ω n,x | x) µ (1)
n (x) = M εh d γ(x)(ε + 1) [1 + r 1,n (x)] (32)
where r 1,n (x) → 0 as n goes to infinity, uniformly in x ∈ Ω. As a consequence, using an elementary Taylor expansion, we get, for all z ∈ B(x, h),

1 γ(z) - q * n,+ (ε, x) nµ (1)
n (x) K h (x -z) -1 = γ(z) 1 + γ(z) γ(x) M ε ε + 1 h d [1 + r 1,n (x)]K h (x -z) + k γ(z) γ(x) M ε ε + 1 h d [1 + r 1,n (x)]K h (x -z)
where k(r)/r → 0 as r goes to 0. Letting

p n (x, z) := γ(z) γ(x) [1 + r 1,n (x)]h d K h (x -z)
and using ( 6), the uniform convergence of r 1,n to 0 and the fact that K is bounded yields

p n (x, z) = h d K h (x -z) + r 2,n (x, z) where sup x∈Ω sup z∈B(x, h) |r 2,n (x, z)| → 0
as n goes to infinity. Especially,

1 γ(z) - q * n,+ (ε, x) nµ (1)
n (x) K h (x -z) -1 = γ(z) 1 + M ε ε + 1 h d K h (x -z) + εr 3,n (ε, x, z) (33)
where r 3,n (ε, x, z) → 0 as ε goes to 0 and n goes to infinity, uniformly in x ∈ Ω and z ∈ B(x, h). Besides, since for every

ε 0 > 0 sup ε<ε0 sup x∈Ω sup z∈B(x, h) q * n,+ (ε, x) nµ (1)
n (x) K h (x -z) - M ε ε + 1 h d K h (x -z) γ(x) → 0
as n goes to infinity and

sup n∈N sup x∈Ω sup z∈B(x, h) M ε ε + 1 h d K h (x -z) γ(x) → 0
as ε goes to 0, (29) yields for ε small enough

sup x∈Ω sup z∈B(x, h) R n q * n,+ (ε, x) nµ (1) n (x) K h (x -z) | x, z → 0 (34)
as n goes to infinity. Using together ( 6), ( 32), ( 33) and (34) entails that there exist functions r such that log ϕ n q * n,+ (ε, x) nµ

(1)

n (x) , x ≤ B(x, h) M ε ε + 1 h d 1 + M ε ε + 1 h d K h (x -z) F (ω n,x | z) K h (x -z) f (z) dz + M ε ε + 1 h d B(x, h) F (ω n,x | z) [r 4,n (x, z) + εr 5,n (ε, x, z)] K h (x -z) f (z) dz.
Recalling ( 5) and ( 23), we get, for n large enough and ε small enough, the inequality

∀ x ∈ Ω, log ϕ n q * n,+ (ε, x) nµ (1) n (x) , x ≤ M ε ε + 1 1 + 2 M ε ε + 1 K 2 2 h d f (x)F (ω n,x | x).
Using this result together with [START_REF] Smith | Extreme value analysis of environmental time series: an application to trend detection in ground-level ozone (with discussion)[END_REF] and recalling that 0 < f ≤ f (x) entails, for n large enough and ε small enough,

∀ x ∈ Ω, u 1,n (x) ≤ exp f -M ε + M ε ε + 1 1 + 2 M ε ε + 1 K 2 2 nh d F (ω n,x | x) .
A straightforward computation shows that M * + := (ε + 1)/(4 K 2 2 ) is the optimal value for M in the above inequality; this value yields

∀ x ∈ Ω, u 1,n (x) ≤ exp - ε 2 8 K 2 2 f nh d F (ω n,x | x) = exp -κ 2 ε 2 nh d F (ω n,x | x)
where κ 2 is a positive constant independent of ε.

Providing a uniform exponential bound for u 2,n (x) starts by noticing that, for all q > 0,

u 2,n (x) ≤ exp -q[ε -1] + n log ϕ n - q nµ (1) n (x)
, x .

Recall [START_REF] Nadaraya | On non-parametric estimates of density functions and regression curves[END_REF] and use the inequality log(1 -r) ≤ -r for all r ∈ (0, 1) to get

log ϕ n (-s, x) ≤ B(x, h) [ψ n (-sK h (x -z) | x, z) -1] f (z) dz.
We choose q as q * n,-(ε, x) :=

ε 4 K 2 2 nh d f (x)F (ω n,x | x)
which, using the ideas developed to control u 1,n (x), yields

∀ x ∈ Ω, u 2,n (x) ≤ exp -κ 2 ε 2 nh d F (ω n,x | x) .
for some constant κ 2 > 0. Setting κ = κ 1 ∧ κ 2 completes the proof of Lemma 3.

The fourth lemma of this section establishes a uniform control of the relative oscillation of x → µ (t) n (x). Before stating this result, we let

m (t) n (x) := E(K 2h (x -X)m (t) n (x, X))
where K := 1l B /V is the uniform kernel on R d , with V being the volume of the unit ball of R d ; let further

K h (u) := h -d K(u/h).
Lemma 4. Assume that (SP ), (K), (A 1 ) and (A 2 ) hold. Pick t ∈ {0, 1} and let ε := ε n be a sequence of positive real numbers such that ε ≤ h. If moreover

• inf x∈Ω ω n,x → ∞ ; • h η sup x∈Ω log ω n,x → 0 ; • sup x∈Ω ∆(log ω n,x )(ε) → 0 ; • sup x∈Ω α(y | x) → 0 as y → ∞ then sup x∈Ω sup z∈B(x, ε) 1 [ε/h] ηK ∨ ∆(log ω n,x )(ε) µ (t) n (z) µ (t) n (x) -1 = O (1) .
Proof of Lemma 4. For all x ∈ Ω and z ∈ B(x, ε), we have

µ (0) n (x) -µ (0) n (z) ≤ E |K h (x -X) -K h (z -X)| 1l {Y >ωn,x} + E K h (z -X) 1l {Y >ωn,x} -1l {Y >ωn,z} =: R (0) 1,n (x, z) + R (0) 2,n (x, z) (35)
and we shall handle both terms in the right-hand side separately. Hypothesis (K) and the inclusion B(z, h) ⊂ B(x, 2h) entail that

|K h (x -X) -K h (z -X)| ≤ M K h d ε h ηK 1l {X∈B(x, 2h)} . (36)
From (36), we get

sup z∈B(x, ε) R (0) 1,n (x, z) ≤ 2 d M K V m (0) n (x) ε h ηK . ( 37
)
Because K is a probability density function on R d with support included in B, applying Lemma 2(iii) implies that

sup x∈Ω m (0) n (x) µ (0) n (x) -1 → 0 as n → ∞ (38)
which, together with (37), yields

sup x∈Ω sup z∈B(x, ε) ε h -ηK R (0) 1,n (x, z) µ (0) n (x) = O(1). ( 39
)
We now turn to the second term. One has

R (0) 2,n (x, z) = E K h (z -X) F (ω n,x | X) -F (ω n,z | X) . (40)
Furthermore, using Lemma 1 with ε ′′ = 0 entails

sup x∈Ω sup x ′ ∈B(x, 2h) sup z∈B(x, ε) 1 ∆(log ω n,x )(ε) F (ω n,z | x ′ ) F (ω n,x | x ′ ) -1 = O(1). (41)
Besides, hypothesis (K) and the inclusion B(z, h) ⊂ B(x, 2h) imply that

E(K h (z -X)m (0) n (x, X)) ≤ 2 d M K V m (0) n (x). ( 42
)
Using the obvious identity and recalling that the support of the random variable K h (z -X) is contained in B(z, h) ⊂ B(x, 2h), ( 40) and (41) yield:

|F (ω n,x | X) -F (ω n,z | X)| = m (0) n (x, X) F (ω n,z | X) F (ω n,x | X) -1 (
sup x∈Ω sup z∈B(x, ε) 1 ∆(log ω n,x )(ε) R (0) 2,n (x, z) m (0) n (x) = O(1), and (38) entails sup x∈Ω sup z∈B(x, ε) 1 ∆(log ω n,x )(ε) R (0) 2,n (x, z) µ (0) n (x) = O(1). ( 44
1 [ε/h] ηK ∨ ∆(log ω n,x )(ε) µ (0) n (z) µ (0) n (x) -1 = O(1)
which shows Lemma 4 in this case.

We now turn to the case t = 1. Note that for all real numbers a, b ≥ 1 such that a = b one has

∀ y ≥ 1, |(log y -log a) + 1l {y>a} -(log y -log b) + 1l {y>b} | ≤ | log b -log a|1l {y>a∧b} . (45)
Inequality (45) then implies, for all x ∈ Ω and z ∈ B(x, ε):

µ (1) n (x) -µ (1) n (z) ≤ E |K h (x -X) -K h (z -X)| (log Y -log ω n,x ) + 1l {Y >ωn,x} + log ω n,x ω n,z E K h (z -X)1l {Y >ωn,x∧ωn,z} =: R (1) 1
,n (x, z) + R (1) 2,n (x, z) (46)
and we shall once again take care of both terms in the right-hand side of this inequality. Start by using (36) to get

sup z∈B(x, ε) R (1) 1,n (x, z) ≤ 2 d M K V m (1) n (x) ε h ηK . (47)
We now use the same idea developed to control R

n (x) µ (1) n (x) -1 → 0 as n → ∞ (1)
which, together with (47), yields

sup x∈Ω sup z∈B(x, ε) ε h -ηK R (1) 1,n (x, z) µ (1) n (x) = O(1). ( 48
)
To control the second term, write

sup z∈B(x, ε) R (1) 2,n (x, z) ≤ ∆(log ω n,x )(ε) sup z∈B(x, ε) E K h (z -X)1l {Y >ωn,x∧ωn,z} .
Note that since ω n,x ∧ ω n,z is either equal to ω n,x or ω n,z , we can write, for all z ∈ B(x, ε)

E K h (z -X)1l {Y >ωn,x∧ωn,z} ≤ E K h (z -X)m (0) n (x, X) ∨ E K h (z -X)m (0) n (z, X) .
Recall now (41) and (43) to obtain, for n large enough, uniformly in x ∈ Ω and z ∈ B(x, ε),

E K h (z -X)1l {Y >ωn,x∧ωn,z} ≤ 2E K h (z -X)m (0) n (x, X) . (49)
Finally, using (42) and (49) yields:

sup x∈Ω sup z∈B(x, ε) 1 ∆(log ω n,x )(ε) R (1) 2,n (x, z) m (0) n (x) = O(1), and (38) entails sup x∈Ω sup z∈B(x, ε) 1 ∆(log ω n,x )(ε) R (1) 2,n (x, z) µ (0) n (x) = O(1) so that Lemma 2(iv) gives sup x∈Ω sup z∈B(x, ε) 1 ∆(log ω n,x )(ε) R (1) 2,n (x, z) µ (1) n (x) = O(1). (50)
Applying (46) together with (48) and (50) implies that

sup x∈Ω sup z∈B(x, ε) 1 [ε/h] ηK ∨ ∆(log ω n,x )(ε) µ (1) n (z) µ (1) n (x) -1 = O(1)
which completes the proof of Lemma 4.

The fifth lemma of this section provides a uniform control of both the difference of two versions of µ (0) n (x) for two families of thresholds that are uniformly asymptotically equivalent and the empirical analogue of this quantity.

Lemma 5. Assume that (SP ), (A 1 ) and (A 2 ) hold. Assume that K is a bounded probability density function on R d with support included in B and that

• inf x∈Ω ω n,x → ∞ ; • h η sup x∈Ω log ω n,x → 0 ; • sup x∈Ω α(y | x) → 0 as y → ∞.

For an arbitrary family of positive sequences

(ρ n,x ) such that sup x∈Ω ρ n,x → 0 as n → ∞, let M n (x) := E(K h (x -X)1l {(1-ρn,x)ωn,x<Y ≤(1+ρn,x)ωn,x} ) and U n (x) := 1 n n i=1 K h (x -X i )1l {(1-ρn,x)ωn,x<Yi≤(1+ρn,x)ωn,x} . Then sup x∈Ω γ(x)M n (x) 2f (x)ρ n,x F (ω n,x | x) -1 → 0
and there exists a positive constant κ such that for all n large enough, one has for every ε > 0 small enough:

∀ x ∈ Ω, P ρ n,x U n (x) M n (x) -1 > ε ≤ 2 exp -κεnh d F (ω n,x | x) .
imsart-aihp ver. 2013/03/06 file: Hill_loiunif_IHP_revised6.tex date: November 21, 2013

Proof of Lemma 5. We start by noting that

M n (x) = E K h (x -X)ρ n,x F (ω n,x | X) F ((1 -ρ n,x )ω n,x | X) ρ n,x F (ω n,x | X) - F ((1 + ρ n,x )ω n,x | X) ρ n,x F (ω n,x | X) .
Use then (SP ) and (A 1 ) to get, for an arbitrary z ∈ B(x, h),

F ((1 ± ρ n,x )ω n,x | z) ρ n,x F (ω n,x | z) = (1 ± ρ n,x ) -1/γ(z) ρ n,x exp (1±ρn,x)ωn,x ωn,x α(v | z) v dv . (51)
Since 0 < γ ≤ γ(z) and sup x∈Ω ρ n,x → 0, a Taylor expansion of the exponential function in a neighborhood of 0 yields

(1 ± ρ n,x ) -1/γ(z) ρ n,x = 1 ρ n,x ∓ 1 γ(z) (1 + r 1,±,n (x, z)) (52)
where r 1,+,n (x, z) and r 1,-,n (x, z) converge to 0 as n → ∞, uniformly in x ∈ Ω and z ∈ B(x, h). Besides, for all u ∈ (-1, 1), We proceed by controlling U n (x). For every x ∈ Ω,

P ρ n,x U n (x) M n (x) -1 > ε = P h d U n (x) -h d M n (x) > ε h d M n (x) ρ n,x .
Notice now that if Z n,i (x) := h d K h (x -X i )1l {(1-ρn,x)ωn,x<Yi≤(1+ρn,x)ωn,x} , then

h d U n (x) -h d M n (x) = 1 n n i=1 [Z n,i (x) -E(Z n,i (x))]
is a mean of bounded, centered, independent and identically distributed random variables. Define

τ n (x) := ε K ∞ nh d M n (x) ρ n,x and λ n (x) := ε K ∞ h d M n (x) ρ n,x
1 Var(Z n, 1 (x)) .

Bernstein's inequality (see Hoeffding [START_REF] Hoeffding | Probability inequalities for sums of bounded random variables[END_REF]) yields, for all ε > 0,

P ρ n,x U n (x) M n (x) -1 > ε ≤ 2 exp - τ n (x)λ n (x) 2(1 + λ n (x)/3)
.

Applying (55) yields, for n large enough,

inf x∈Ω τ n (x) nh d F (ω n,x | x) ≥ εf γ K ∞ . (56)
Moreover, since Z 2 n, 1 (x) ≤ K ∞ Z n, 1 (x), it follows that sup Using (56) and (57) it is then clear that, for all n large enough, if ε > 0 is small enough, there exists a positive constant κ that is independent of ε such that

∀ x ∈ Ω, P ρ n,x U n (x) M n (x) -1 > ε ≤ 2 exp -κεnh d F (ω n,x | x) .
This completes the proof of Lemma 5.

The final lemma is the last step in the proof of Theorem 2.

Lemma 6. Let (X n ) be a sequence of positive real-valued random variables such that for every positive nonrandom sequence (δ n ) converging to 0, the random sequence (δ n X n ) converges to 0 almost surely. Then Hence (δ n X n ) does not converge almost surely to 0, from which the result follows.

  imsart-aihp ver. 2013/03/06 file: Hill_loiunif_IHP_revised6.tex date: November 21, 2013 Proof of Theorem 2. Note that because nh d → ∞, the hypothesis lim sup n→∞ sup x∈Ω v n (x)∆(log ω n,x )(h) < ∞ entails condition (C). Besides,

  imsart-aihp ver. 2013/03/06 file: Hill_loiunif_IHP_revised6.tex date:[START_REF] Hoeffding | Probability inequalities for sums of bounded random variables[END_REF]

  4,n = r 4,n (x, z) and r 5,n = r 5,n (ε, x, z) satisfying sup x∈Ω sup z∈B(x, h) |r 4,n (x, z)| → 0 as n → ∞ and sup x∈Ω sup z∈B(x, h) |r 5,n (ε, x, z)| → 0 as ε → 0 and n → ∞ imsart-aihp ver. 2013/03/06 file: Hill_loiunif_IHP_revised6.tex date: November 21, 2013

  (x, z): applying Lemma 2(iii) entails sup x∈Ω m

( 1 + 2 FE

 12 n,x | z)| log(1 + u)| if u > 0 α((1 + u)ω n,x | z)| log(1 + u)| if u < 0 (53) so that, because inf x∈Ω ω n,x → ∞, sup x∈Ω ρ n,x → 0 and sup x∈Ω α(y | x) → 0 as y → ∞: r 2,±,n (x, z))where r 2,+,n (x, z) and r 2,-,n (x, z) converge to 0 as n → ∞, uniformly in x ∈ Ω and z ∈ B(x, h). x)ωn,xωn,x α(v | z) v dv → 0 as n → ∞.Plugging this together with (52) into (51) and recalling that 0< γ ≤ γ(z) ((1 -ρ n,x )ω n,x | z) ρ n,x F (ω n,x | z) -F ((1 + ρ n,x )ω n,x | z) ρ n,x F (ω n,x | z) -1 → 0 as n → ∞. Consequently, sup x∈Ω M n (x) 2E K h (x -X)ρ n,x F (ω n,x | X)/γ(X) -1 → 0 as n → ∞. K h (x -X)F (ω n,x | X)/γ(X) µ (0) n (x)/γ(x) -1 → 0.It only remains to recall (54) and to apply Lemma 2(iv) to obtain sup x∈Ω γ(x)M n (x) 2f (x)ρ n,x F (ω n,x | x) -1 → 0. (55)

1

 1 (x)) ε K ∞ h d M n (x) n → ∞. Finally, it holds that τ n (x)λ n (x) 2(1 + λ n (x)/3) ≥ inf x∈Ω τ n (x) nh d F (ω n,x | x) inf x∈Ω 1 2(1/λ n (x) + 1/3) nh d F (ω n,x | x).

P 2 .

 2 lim sup n→∞ X n = +∞ = 0 i.e. X n = O(1) almost surely. Proof of Lemma 6. Assume that there exists ε > 0 such that P lim sup n→∞ X n = +∞ ≥ ε. Since by definition lim sup n→∞ X n = lim n→∞ sup p≥n X p is the limit of a nonincreasing sequence, one has∀ k ∈ N, ∀ n ∈ N, P   p≥n {X p ≥ k}   ≥ ε ⇒ ∀ k ∈ N, ∀ n ∈ N, ∃ n ′ ≥ n, P   n ′ p=n {X p ≥ k}   ≥ ε/2.imsart-aihp ver. 2013/03/06 file: Hill_loiunif_IHP_revised6.tex date: November 21, 2013It is thus easy to build an increasing sequence of integers (N k ) such that∀ k ≥ 1, Let δ n = 1/k if N k ≤ n < N k+1 . It is clear that (δ n ) isa positive sequence which converges to 0. Besides, for all k ∈ N \ {0} it holds that P sup p≥N k δ p X p ≥ 1 = P

imsart-aihp ver. 2013/03/06 file: Hill_loiunif_IHP_revised6.tex date:[START_REF] Hoeffding | Probability inequalities for sums of bounded random variables[END_REF]

Acknowledgements

The authors acknowledge the editor and an anonymous referee for their helpful comments which led to significant enhancements of this article.