Randolf Altmeyer

Ronan Le Guével
email: ronan.leguevel@univ-rennes2.fr

Optimal L 2 -approximation of occupation and local times for symmetric stable processes

Keywords: occupation time, local time, stable process, Lévy process, lower bound

The L 2 -approximation of occupation and local times of a symmetric α-stable Lévy process from high frequency discrete time observations is studied. The standard Riemann sum estimators are shown to be asymptotically efficient when 0 < α ≤ 1, but only rate optimal for 1 < α ≤ 2. For this, the exact convergence of the L 2 -approximation error is proven with explicit constants.

Introduction

Let X = (X t ) t≥0 be a scalar stochastic process. Two path dependent functionals of X which are of interest in many applications are its occupation and local times respectively defined by

O T (A) = T 0 1 A (X t )dt and L T (y) = dO T dy (y), (1)
which measure the time the process spends inside a Borel set A ⊆ R or at a point y ∈ R, whenever the occupation measure A → O T (A) is absolutely continuous with respect to the Lebesgue measure. We aim at studying optimal L 2 -approximations of these functionals given the observations X t k at t k = k∆ n for k = 1, . . . , n with time distance ∆ n = T /n, where the time horizon T > 0 is fixed and in the high frequency limit as n → ∞. The minimal L 2 -error is achieved for the conditional expectations

E[O T (A)|G n ] and E[L T (y)|G n ]
, where G n is the sigma field generated by the X t k , but these two conditional expectations may be unfeasible to compute when the law of X 1 is unknown. Instead, the standard estimators in the literature are based on integral approximations using the Riemann sums ÔT,n (A) = ∆ n n k=1

1 A (X t k-1 ) and LT,n (y) = ∆ n 2h n n k=1

1 [y-hn,y+hn] (X t k-1 ) (2)
for some bandwidth parameter h n > 0. These approximations may be far from optimality since they crucially depend on the smoothness of the law of the underlying process.

The main result of this article is to settle this question in the context of symmetric αstable processes by proving exact convergence results for the L 2 -approximation errors.

The approximation of occupation and local times is important in many applications and has been extensively studied in the literature. For stationary continuous time stochastic processes for instance, the irregularity of the sample paths implies non-standard rates of convergence in the non-parametric estimation of the probability density with kernel type estimators, as has been noticed in [START_REF] Castellana | On smoothed probability density estimation for stationary processes[END_REF][START_REF] Bosq | Nonparametric statistics for stochastic processes[END_REF]. The question of optimality with respect to the sampling of discrete time observation schemes has been studied in [START_REF] Blanke | Optimal sampling for density estimation in continuous time[END_REF], while the rate optimality is considered in [START_REF] Comte | Super optimal rates for nonparametric density estimation via projection estimators[END_REF] through the study of a projection estimator. We focus in this article on non-parametric methods, but for processes which are no longer stationary. For scalar diffusion processes X and intervals A, the standard estimators have been studied by several authors [START_REF] Borodin | On the character of convergence to Brownian local time[END_REF][START_REF] Ogawa | On the discrete approximation of occupation time of diffusion processes[END_REF][START_REF] Jacod | Rates of convergence to the local time of a diffusion[END_REF][START_REF] Ngo | Approximations of nonsmooth integral type functionals of one dimensional diffusion processes[END_REF], satisfying the rates of convergence ∆ n for L T (y). These rates can be explained in the context of L 2 -approximations of integral functionals T 0 g(X t )dt for non-smooth integrands g [START_REF] Altmeyer | Estimation error for occupation time functionals of stationary Markov processes[END_REF]. In this way, [START_REF] Altmeyer | Central limit theorems for discretized occupation time functionals[END_REF][START_REF] Altmeyer | Approximation of occupation time functionals[END_REF] obtain similar results for more general Markovian and non-Markovian processes such as semimartingales or fractional Brownian motion. Rate optimality of the Riemann sum estimators in the case of Brownian motion (with drift) can be obtained from [START_REF] Ogawa | On the discrete approximation of occupation time of diffusion processes[END_REF][START_REF] Ivanovs | Optimal estimation of some random quantities of a Lévy process[END_REF][START_REF] Altmeyer | Approximation of occupation time functionals[END_REF], but it is unclear if their methods extend to jump processes, or if Riemann estimators are asymptotically efficient in the sense of reaching the minimal asymptotic error. More recently, there is also some interest in numerical analysis for the L p -approximation error in the context of analysing Euler schemes with non-degenerate coefficients ( [START_REF] Neuenkirch | The Euler-Maruyama scheme for SDEs with irregular drift: Convergence rates via reduction to a quadrature problem[END_REF], [START_REF] Müller-Gronbach | Sharp lower error bounds for strong approximation of sdes with discontinuous drift coefficient by coupling of noise[END_REF]), see also [START_REF] Butkovsky | Approximation of sdes: a stochastic sewing approach[END_REF].

Similar to [START_REF] Ivanovs | Optimal estimation of some random quantities of a Lévy process[END_REF][START_REF] Altmeyer | Approximation of occupation time functionals[END_REF], we assess the question of optimality by studying the conditional expectations

E[O T (A)|G n ], E[L T (y)|G n ].
For explicit computations, we restrict to symmetric α-stable processes for 0 < α ≤ 2, but we expect that our results hold also for more general Lévy processes. We prove the exact constants for the asymptotic L 2 -approximation errors of the conditional expectations and for the Riemann sum estimators. In both cases we obtain the rates of convergence ∆

(1+min(1,1/α))/2 n
for occupation times (up to log-factors) and ∆

(1-1/α)/2 n for local times. In particular, we show that the Riemann sum estimators are rate optimal, but asymptotically efficient only for α ≤ 1, surprisingly. Let us point out that while the conditional expectations are explicit estimators up to the possibly unknown parameter α, they depend on the marginal densities of X and are therefore not known analytically for α < 2, requiring numerical approximations. Our results imply, however, that it is sufficient to use the standard estimators. The general proof strategy is to compute L 2 terms explicitly, leading to Riemann inte-grals and then argue by dominated convergence or improper integral divergence, using precise asymptotics with respect to the law of α-stable distributions.

The paper is organised as follows. In Section 2.1 we recall properties of stable processes. Section 2.2 presents the L 2 -approximation results for standard estimators of occupation and local times. Consistency and rates of convergence in a general setting are discussed, along with exact asymptotics for some important cases. Section 2.3 compares these results to the exact asymptotics for the conditional expectations. All proofs are deferred to Section 3.

Main results

α-stable processes

Suppose that X = (X t ) t≥0 is a scalar symmetric α-stable Lévy process for 0 < α ≤ 2, that is X 0 = 0 and X is a self-similar Lévy process such that

(X bt ) t≥0 d = b 1/α (X t ) t≥0 , b > 0.
In particular, each X t has for t > 0 the characteristic function u → E[e iuXt ] = e -|u| α t and thus the Lebesgue density

f α,t (x) = 1 t 1/α f α x t 1/α with f α (x) = 1 π ∞ 0 e -t α cos(xt)dt, x ∈ R, (3)
cf. [START_REF] Sato | Lévy Processes and Infinitely Divisible Distributions[END_REF]. For α = 2, X is a Brownian motion (up to scaling factor), and a Cauchy-process for α = 1. Since X has right-continuous paths, the occupation time O T (A) in ( 1) is well-defined for each Borel set A ⊆ R and any 0 < α ≤ 2. We write

O T (y) = O T ([y, ∞)), y ∈ R.
The local time process L T (y) in (1), however, exists only for α > 1, cf. [START_REF] Khoshnevisan | Local times of additive Lévy processes[END_REF]Theorem 2.1]. Recall also the occupation time formula

T 0 f (X s )ds = R f (x)L T (y)dy, (4)
which holds for all nonnegative measurable functions f , cf. [START_REF] Samorodnitsky | Stochastic Processes and Long Range Dependence[END_REF].

Results for the Riemann estimators

We begin by a simple, but general consistency result for ÔT,n (A), which can be shown exactly as in [START_REF] Ogawa | On the discrete approximation of occupation time of diffusion processes[END_REF]Proposition 2.1]. Convergence of LT,n (y) to L T (y) in probability (and in L 2 ) will follow from Theorem 4 below.

Proposition 1. Let X be a stochastic process with right-continuous (or left-continuous) paths such that for all t > 0 the law of X t is absolutely continuous with respect to the Lebesgue measure λ. If A ⊆ R is a Borel set such that λ( Ā\ Å) = 0, where Ā and Å denote the closure and the interior of A, then P-almost surely

lim n→∞ ÔT,n (A) = O T (A).
Let us consider L 2 -rates of convergence of this estimator. For α > 1 we follow the proof strategy outlined in [START_REF] Altmeyer | Approximation of occupation time functionals[END_REF]Corollaries 6 and 7] for fractional Brownian motion by upper bounding the characteristic function of the bivariate distributions (X t , X t ) for 0 < t < t ≤ T . This leads to a general control of the error approximation on every interval [a, b].

Theorem 2. Let 1 < α ≤ 2 and let -∞ ≤ a ≤ b ≤ ∞. Then for sufficiently small ε > 0 ∆ -1-1/α n ÔT,n ([a, b]) -O T ([a, b]) 2 L 2 (P) ≤ C(1 ∨ T -ε )T 1-1/α , with C < ∞ is independent of a, b, T and n.
The proof of Theorem 2 breaks down for 0 < α ≤ 1 due to the singularity of f α,t near t = 0, yielding only suboptimal rates of convergence. This can be resolved by assuming an initial distribution X 0 having a bounded Lebesgue density and the same upper bound from Theorem 2 still applies up to a small polynomial loss in the rate of convergence, as has been shown in [START_REF] Altmeyer | Approximation of occupation time functionals[END_REF]Theorem 15], noting that indicator functions of bounded intervals have fractional Sobolev regularity s < 1/2.

In the important case when A = [0, ∞) is a half-line, we will now obtain exact convergence results with explicit constants for all 0 < α ≤ 2. This is new even in the Brownian case (upper and lower bounds in the this case can be found in [START_REF] Ogawa | On the discrete approximation of occupation time of diffusion processes[END_REF]Proposition 2.3]).

Theorem 3. Define ψ(x) := (x -x ) -(x -x ) 2 , x ≥ 0, where x is the integer part of x. If 1 < α ≤ 2, then lim n→∞ ∆ -1-1/α n ÔT,n (0) -O T (0) 2 L 2 (P) = T 1-1/α 2Γ(1/α) πα 2 E[|X 1 |] ∞ 0 ψ(x) x 2-1/α dx. If 0 < α < 1, then lim n→∞ ∆ -2 n (log n) -1 ÔT,n (0) -O T (0) 2 L 2 (P) = Γ(α) sin( πα 2 ) 12π E[|X 1 | -α ]. If α = 1, then lim n→∞ ∆ -2 n (log n) -2 ÔT,n (0) -O T (0) 2 L 2 (P) = 1 12π 2 .
We will see in the next section that the additional log n factors for 0 < α ≤ 1 are necessary. Using Theorem 2 we also state a general upper bound for the estimation of the local time.

LT,n (y) -L T (y) 2 L 2 (P) ≤ C(1 ∨ T -ε )T 1-1/α (h α-1 n + ∆ 1+1/α n h -2 n ),
where C < ∞ is independent of a, b, T and n. If

h n = ∆ 1/α n , then ∆ 1/α-1 n LT,n (y) -L T (y) 2 L 2 (P) ≤ C(1 ∨ T -ε )T 1-1/α .
In the Brownian case we recover the rate of convergence ∆ 1/4

n from [START_REF] Jacod | Rates of convergence to the local time of a diffusion[END_REF], and therefore improve on [2, Corollary 7] and [START_REF] Ngo | Approximations of nonsmooth integral type functionals of one dimensional diffusion processes[END_REF]Theorem 2.6].

Optimal estimation results

In this section we will derive the exact asymptotics for the L 2 -error of the conditional expectations E[O T (y)|G n ] and E[L T (y)|G n ] as n → ∞ with explicit constants. Note that local times are square integrable and therefore E[L T (y)|G n ] is well-defined (see [START_REF] Marcus | Lp moduli of continuity of Gaussian processes and local times of symmetric Lévy processes[END_REF]).

Theorem 5. If 1 < α ≤ 2 and y ∈ R, then lim n→∞ ∆ -1-1/α n E[O T (y)|G n ] -O T (y) 2 L 2 (P) = 2E[L T (y)] ∞ 0 E [Var ( O 1 (x)| X 1 )] dx. If 0 < α < 1, then lim n→∞ ∆ -2 n (log n) -1 E[O T (0)|G n ] -O T (0) 2 L 2 (P) = Γ(α) sin( πα 2 ) 12π E[|X 1 | -α ]. If α = 1, then lim n→∞ ∆ -2 n (log n) -2 E[O T (0)|G n ] -O T (0) 2 L 2 (P) = 1 12π 2 .
In view of Theorems 2 and 3 we conclude that the Riemann estimator ÔT,n (y) is rate optimal for all 1 < α ≤ 2 and all y ∈ R, while for 0 < α ≤ 1, ÔT,n (0) is even asymptotically efficient and achieves the minimal possible error. In particular, the Riemann estimator automatically recovers the different regimes for α. Efficiency does not hold true for 1 < α ≤ 2, in particular not for Brownian motion, as the next proposition shows. Proposition 6. For all 1 < α ≤ 2 we have We therefore conclude that the Riemann estimator ÔT,n (0) always has a strictly larger estimation error than E[O T (0)|G n ] for all 1 < α ≤ 2. To get some idea of how large C α is, let us use ( 7) from the proof of Proposition 6 to lower bound C α by

lim n→∞ ∆ -1-1/α n ÔT,n (0) -O T (0) 2 L 2 (P) lim n→∞ ∆ -1-1/α n E[O T (0)|G n ] -O T (0) 2 L 2 (P) =: C α > 1.
C α > Cα := 2(2α + 1)(α -1) α 3 ∞ 0 ψ(x) x 2-1/α dx.
Now Cα can be easily evaluated numerically, and Figure 1 shows that α → Cα increases with α. In particular, C2 ≈ 2.08. We conclude by an exact convergence result for local times at any y ∈ R, which proves together with Theorem 4 the rate optimality of the Riemann estimator LT,n (y). We conjecture that there is an analogous statement to Proposition 6 for local times, but a proof seems difficult. (i) Section 3 of [START_REF] Altmeyer | Central limit theorems for discretized occupation time functionals[END_REF] shows that the Riemann estimator is asymptotically efficient as n → ∞ for approximating integral functionals with smooth integrands when X is a Brownian motion, implying that the integral approximations in (1) can not be improved asymptotically by considering higher order quadrature rules such as the trapezoidal rule.

Theorem 7. Let 1 < α ≤ 2, y ∈ R and set C(α) = -(α -1)Γ(α) cos( πα 2 ). Then lim n→∞ ∆ 1/α-1 n E[ L T (y)| G n ] -L T (y) 2 L 2 (P) = E[L T (y)] C(α) 2 R E[Var( L 1 (x)| X 1 )]dx.
(ii) From ( 14) in the proof of Lemma 16, we can see that

E[O T (A)|G n ] = n k=1 g α (X t k-1 , X t k -X t k-1 ), A ⊆ R, (5)
for a function g α depending explicitly on α and the density f α . This suggests that an asymptotically efficient estimator for 1 < α ≤ 2 needs to use also the independent increments X t k -X t k-1 besides the time points X t k-1 , but this seems not to be necessary for 0 < α ≤ 1.

(iii) The identity [START_REF] Blanke | Optimal sampling for density estimation in continuous time[END_REF] suggests to use the conditional expectation as estimator. While it is possible to pre-estimate α from the observations, cf. [START_REF] Belomestny | Spectral estimation of the fractional order of a Lévy process[END_REF], f α is generally not known in closed form and needs to be numerically approximated. In view of the good performance of the Riemann estimators across different models, cf. [START_REF] Altmeyer | Approximation of occupation time functionals[END_REF], one should refrain from using (5) as estimator if X is not precisely a symmetric αstable process.

Proofs

All along the proofs, Z, Z are generic standard α-stable random variables, independent of each other. For two non-negative functions f and g, we write f g if sup f g < +∞.

3.1 Proofs of results for the Riemann estimators

O T (0) 2 L 2 (P) + ÔT,n (0) 2 L 2 (P) -2E[O T (0) ÔT,n (0)]. (6)
We first compute these three expression explicitly. For this, observe the following properties of marginal and joint densities of the standard α-stable distribution.

Lemma 10. Define

ϕ(x, y) = P(Z ≥ 0, Z ≥ 0, x 1/α Z ≤ y 1/α Z), x, y ≥ 0,
Then we have for 0 < r < t that P(X r ≥ 0, X t ≥ 0) = 1 4 + ϕ(t -r, r). Proof. Observe for 0 < r < t by symmetry and independence of increments that P(X r ≥ 0, X t -X r ≥ 0) = 1 4 . Therefore

P(X r ≥ 0, X t ≥ 0) = 1 4 + P(X r ≥ 0, 0 ≥ X t -X r ≥ -X r ).
By the 1/α-self-similarity we can write

X t -X r d = (t -r) 1/α Z, X r d = r 1/α Z. Symmetry yields Z d = -Z such that P(X r ≥ 0, X t ≥ 0) = 1 4 + P( Z ≥ 0, 0 ≤ (t -r) 1/α Z ≤ r 1/α Z) = 1 4 + ϕ(t -r, r).
Lemma 11. The function ϕ from Lemma 10 enjoys the following properties for x, y, b > 0, 0 < a ≤ T :

(i) ϕ(x, y) + ϕ(y, x) = 1 4 , (ii) ϕ(ax, ay) = ϕ(x, y), (iii) a 0 ϕ(a -x, x)dx = a 8 , (iv)
T a ϕ(x -a, a)dx = 1 4 E[min(T, aD -1 )] -a 4 with D = (1 + | Z/Z| α ) -1 . Proof. (i) follows from (Z, Z) d = ( Z, Z), (ii) is clear. For (iii), substitution shows a 0 ϕ(x, a -x)dx = a 0 ϕ(a -x, x
)dx and the claim follows from (i) and

a 0 ϕ(x, a -x)dx = 1 2 ( a 0 ϕ(x, a -x)dx + a 0 ϕ(a -x, x)dx).
For (iv), on the other hand, we have

T a ϕ(x -a, a)dx = E[1 {Z≥0} 1 { Z≥0} T a 1 {(x-a)|Z| α ≤a| Z| α } dx] = E[1 {Z≥0} 1 { Z≥0} T a 1 {x≤a |Z| α +| Z| α |Z| α } dx] = 1 4 E[min(T, a(1 + | Z/Z| α ))] - a 4 .
Lemma 12. We have O T (0) 2 L 2 (P) = 3 8 T 2 . Proof. Use Lemma 10 to obtain

O T (0) 2 L 2 (P) = 2 T 0 T r P(X r ≥ 0, X t ≥ 0)dtdr = 2 T 0 T r 1 4 dtdr + 2 T 0 T -r 0 ϕ(x, r)dxdr.
Consequently, by changing the variables x, r and using Lemma 11(i)

O T (0) 2 L 2 (P) = T 2 4 + T 0 T -r 0 ϕ(x, r)dxdr + T 0 T -x 0 ϕ(x, r)drdx = T 2 4 + T 0 T -r 0 (ϕ(x, r) + ϕ(r, x))dxdr = T 2 4 + T 0 T -r 0 1 4 dxdr = 3 8 T 2 .
Lemma 13. We have ÔT,n (0

) 2 L 2 (P) = 3 8 T 2 + 3 8 T ∆ n + 1 4 ∆ 2 n .
Proof. Since X 0 = 0, expanding the square shows that ÔT,n (0) 2 L 2 (P) equals

E[(∆ n + ∆ n n-1 k=1 1 [0,∞) (X t k )) 2 ] = ∆ 2 n + 3∆ 2 n n-1 k=1 P(X t k ≥ 0) + 2∆ 2 n n-2 k=1 n-1 j=k+1 P(X t k ≥ 0, X j∆n ≥ 0).
As the distribution of X t is symmetric for t > 0, the first two terms are just ∆

2 n + 3 2 ∆ 2 n (n -1)
. Together with Lemma 10, the last display is thus equal to

∆ 2 n + 3 2 ∆ 2 n (n -1) + 2∆ 2 n n-2 k=1 n-1 j=k+1 1 4 + 2∆ 2 n n-2 k=1 n-1 j=k+1 ϕ((j -k)∆ n , t k ).
By Lemma 11(ii) and an index change in the last sum we have

2 n-2 k=1 n-1 j=k+1 ϕ((j -k)∆ n , t k ) = 2 n-2 k=1 n-k-1 j=1 ϕ(j, k) = n-2 k=1 n-k-1 j=1 (ϕ(j, k) + ϕ(k, j)). Lemma 11(i) thus implies ÔT,n (0) 2 L 2 (P) = ∆ 2 n 3n -1 2 + 2∆ 2 n n-2 k=1 n-1 j=k+1 1 4 + ∆ 2 n n-2 k=1 n-k-1 j=1 1 4 = 3 8 T 2 + 3 8 T ∆ n + 1 4 ∆ 2 n .
Lemma 14. We have

2E[O T (0) ÔT,n (0)] = 3 4 T 2 + 3 8 T ∆ n -1 8 ∆ 2 n E[ ψ(nD) D(1-D) ] with D = (1 + | Z/Z| α ) -1 . Proof. Let A = 2E[O T (0) ÔT,n (0)
]. Since X 0 = 0, Lemma 10 and splitting the integral into two parts show

A = 2∆ n T 0 P(X r ≥ 0)dr + 2∆ n n-1 k=1 T 0 P(X t k ≥ 0, X r ≥ 0)dr = T ∆ n + 2∆ n n-1 k=1 T 0 1 4 dr + 2∆ n n-1 k=1 t k 0 ϕ(t k -r, r)dr + 2∆ n n-1 k=1 T t k ϕ(r -t k , t k )dr.
We thus get from Lemma 11(iii), (iv) that

A = T ∆ n + T ∆ n 2 (n -1) + ∆ n n-1 k=1 t k 4 + ∆ n 2 ( n-1 k=1 E[min(T, k D ∆ n )] - n-1 k=1 t k ). Since min(T, k D ∆ n ) = T min(1, k nD ), this means that n-1 k=1 min(T, k D ∆ n ) = T n-1 k=1 min(1, k nD ) = T nD k=1 k nD + T (n -1 -nD ) = T nD ( nD + 1) 2nD -T nD + T (n -1) = T 2 (1 - ψ(nD) nD -nD) + T (n -1).
Noting that

D d = 1 -D, we have E[D] = 1/2.
The last line therefore has expectation

- T 2 - ∆ n 2 E ψ(nD) D + 3T n 4 = - T 2 - ∆ n 4 E ψ(nD) D(1 -D) + 3T n 4 ,
where we used for the last equality the relation ψ(nx) = ψ(n(1 -x)), which follows from x + n -x = n -1. From this obtain the result.

Proof of Theorem 3. The decomposition [START_REF] Borodin | On the character of convergence to Brownian local time[END_REF] and Lemmas 12, 13, 14 show

V T,n = ∆ 2 n 4 + ∆ 2 n 8 E ψ(nD) D(1 -D) .
Denote the Lebesgue density of

D = (1 + | Z/Z| α ) -1 by f D . By the symmetry D d = 1 -D, we have f D (x) = f D (1 -x).
A change of variables combined with the equality ψ(nx) = ψ(n(1 -x)) thus allows for rewriting the last display as

V T,n = ∆ 2 n 4 + ∆ 2 n 4 n/2 0 ψ(x) x(1 -x/n) f D x n dx.
The result follows from studying the dx-integral integral as n → ∞ for different α. For 1 < α ≤ 2, we find from Lemma 19(i) and 0 < x < n/2 that

n -1+1/α ψ(x) x(1 -x/n) f D x n ψ(x) x 2-1/α ,
which is integrable for x > 0. Part (i) of the theorem follows then immediately from the dominated convergence theorem and the convergence in Lemma 19(i). For part (ii) and 0 < α < 1, note that f D is bounded and the limit f D (0+) := lim x→0 f D (x) exists according to Lemma 19(ii). By Lemma 20(i), we conclude that

1 log n n/2 0 ψ(x) x(1 -x/n) f D x n dx = f D (0+) log n n/2 0 ψ(x) x(1 -x/n) dx + 1 log n n/2 0 ψ(x) x(1 -x/n) (f D x n -f D (0+))dx = f D (0+) 6 + o(1).
Finally, for part (iii) and α = 1, we find from Lemmas 19(iii) and 20(ii) that

1 (log n) 2 n/2 0 ψ(x) x(1 -x/n) f D x n dx = 1 (log n) 2 4 π 2 n/2 0 ψ(x) log((x/n) -1 -1) x(1 -x/n)(1 -2x/n) dx
converges to 1 3π 2 , thereby implying the claimed convergence. This finishes the proof.

Proof of Proposition 6

According to Theorems 5 and 3 for 1 < α ≤ 2 it suffices to show

2T 1-1/α Γ(1/α) πα 2 E[|Z|] ∞ 0 ψ(x) x 2-1/α dx 2E[L T (0)] ∞ 0 E [Var ( O 1 (x)| Z)] dx > 1.
From (3) and the occupation time formula (4) we infer

E[L T (0)] = T 0 f α,t (0)dt = Γ(1/α) π(α-1) T 1-1/α . The result follows from ∞ 0 E [Var ( O 1 (x)| Z)] dx < α 2(2α + 1) E[|Z|], (7)
∞ 0 ψ(x) x 2-1/α dx ≥ α 3 2(α -1)(2α + 1) . (8)
Recall [START_REF] Ngo | Approximations of nonsmooth integral type functionals of one dimensional diffusion processes[END_REF] in Lemma 18 with ρ r,s being the density of X r ∧ X s such that

∞ 0 E [Var ( O 1 (x)| Z)] dx < ∞ 0 E 1 0 1 Xr≥x dr 2 dx = ∞ 0 1 0 1 r P(X r ∧ X s ≥ x)dsdrdx = 1 0 1 r ∞ 0 uρ r,s (u)dudsdr = α 2 2(α + 1)(2α + 1) ∞ 0 uf α (u)du + 1 0 1 r ∞ 0 ∞ 0 uf α,r (u + v)f α,s-r (v)dvdudsdr = α 2 4(α + 1)(2α + 1) E[|Z|] + 1 0 1 r E[X s 1 Xs≥0 1 Xr≤0 ]dsdr.
By the symmetry X d = -X we deduce for s > r

0 = E[X s (1 Xs≥0 1 Xr≤0 + 1 Xs≤0 1 Xr≥0 )] = E[X s (21 Xs≥0 1 Xr≤0 + 1 -1 Xs≥0 -1 Xr≤0 )]
and therefore

2E[X s 1 Xs≥0 1 Xr≤0 ] = E[X s 1 Xs≥0 ] -E[X s 1 Xr≤0 ] = E[X s 1 Xs≥0 ] -E[(X s -X r )1 Xr≤0 ] -E[X r 1 Xr≤0 ] = s 1/α -r 1/α 2 E[|Z|].
From this obtain [START_REF] Bosq | Nonparametric statistics for stochastic processes[END_REF]. To conclude let us compute

∞ 0 ψ(x) x 2-1/α dx = 1 0 x-x 2 x 2-1/α dx + ∞ k=1 1 0 x-x 2 (x+k) 2-1/α dx. Integration by parts entails ∞ 0 ψ(x) x 2-1/α dx = α 2 α + 1 + α 2 α -1 ∞ k=1 2α α + 1 ((1 + k) 1+1/α -k 1+1/α ) -k 1/α -(1 + k) 1/α = α 2 α + 1 + α 2 α -1 ∞ k=1 2α α + 1 k 1+1/α [(1 + 1 k ) 1+1/α -1] -k 1/α -k 1/α (1 + 1 k ) 1/α .
We then use the inequalities

(1 + x) α -1 ≥ α x + α (α -1) 2 x 2 - α (α -1)(2 -α ) 6 x 3
for α ∈ (1, 2) and x ∈ (0, 1), as well as

(1 + x) α ≤ 1 + α x - α (1 -α ) 2 x 2 + α (1 -α )(2 -α ) 6 x 3
for α ∈ (0, 1) and x ∈ (0, 1) to obtain

∞ 0 ψ(x) x 2-1/α dx ≥ α 2 α + 1 + α 2 α -1 ∞ k=1 [ 2α α + 1 k 1+1/α [ α + 1 αk + α + 1 2α 2 k 2 - (α + 1)(α -1) 6α 3 k 3 ] -k 1/α -k 1/α (1 + 1 αk - α -1 2α 2 k 2 + (α -1)(2α -1) 6α 3 k 3 )] = α 2 α + 1 + 1 6 ∞ k=1 k 1/α-2 - (2α -1) 6α ∞ k=1 k 1/α-3 ≥ α 2 α + 1 + 1 6 ∞ 1 x 1/α-2 dx - (2α -1) 6α ∞ 0 x 1/α-3 dx = 6α 3 -6α 2 + α + 1 6(α -1)(α + 1) > α 3 2(α -1)(2α + 1)
, which holds for α > 1. This yields [START_REF] Butkovsky | Approximation of sdes: a stochastic sewing approach[END_REF] and finishes the proof.

Proof of Theorem 2

Let g = 1 [a,b] , 0 ≤ t, t ≤ T and define

E t,t := E[(g(X t ) -g(X ∆n t/∆n ))(g(X t ) -g(X ∆n t /∆n ))].
The main idea of the proof is to get a tight control on E t,t using Fourier calculus, see Lemma

O T ([a, b]) -ÔT,n ([a, b]) 2 L 2 (P) ≤ 2∆ 2 n + 4(A 1 + A 2 ), with A 1 = n k=2 n k =k+1 t k t k-1 t k t k -1 E t,t dt dt, A 2 = n k=2 t k t k-1 t k t E t,t dt dt.
Lemma 15(i,ii) gives

|A 1 | (1 ∨ T ε/α )∆ 2 n n k=2 (t -(1+ε)/α k-1 ∆ 1/α n + t -1-ε/α k-1 ∆ n log n) (1 ∨ T ε/α )(T 1-1/α-ε/α ∆ 1+1/α n + ∆ 2-ε/α n log n) (1 ∨ T ε/α )T 1-1/α-ε/α ∆ 1+1/α n , |A 2 | ∆ 2 n n k=2 (∆ 1/α n (t -1/α k-1 + t -(1+ε)/α k-1 ) + ∆ 1-ε/α n t -1-ε/α k-1 + ∆ 1-2ε/α n t -1 k-1 ) (1 ∨ T 2ε/α )T 1-1/α-2ε/α ∆ 1+1/α n ,
using that n 1/α+ε/α-1 log n ≤ 1 for α > 1 and ε small enough. The result follows from modifying ε.

Lemma 15. The following holds:

(i) If t k-1 ≤ t < t k and t k -1 ≤ t < t k , k = k , then for sufficiently small ε > 0 |E t,t | (1 ∨ T ε/α )(t -(1+ε)/α k-1 ∆ 1/α n + t -1-ε/α k-1 ∆ n log n).
(ii) If t k-1 < t < t < t k , then for sufficiently small ε > 0

|E t,t | ∆ 1/α n (t -1/α k-1 + t -(1+ε)/α k-1 ) + ∆ 1-ε/α n t -1-ε/α k-1 + ∆ 1-2ε/α n t -1 k-1 . Proof. Assume first that -∞ < a < b < ∞. The Plancherel theorem (on R 2 ) shows for 0 < t < t ≤ T that E[g(X t )g(X t )] = (2π) -2 R 2 Fg(u)Fg(v)ϕ(-u, t; -v, t )d(u, v), (9)
where ϕ(u, t; v, t ) = E[e iuXt+ivX t ] is the characteristic function of (X t , X t ) and Fg(u) = R g(x)e iux dx = (iu) -1 (e iub -e iua ) is the Fourier transform of g. By independence of increments, we have ϕ(u, t; v, t ) = e -|u+v| α t-|v| α (t -t) for t > t. From ( 9)

|E t,t | R 2 (1 ∧ |u| -1 )(1 ∧ |v| -1 )|E -u,-v t,t |d(u, v), (10)
where

E u,v t,t = ϕ(u, t; v, t ) -ϕ(u, t; v, t k -1 ) -ϕ(u, t k-1 ; v, t ) + ϕ(u, t k-1 ; v, t k -1 )
. By an approximation argument, this upper bound is also true for any

-∞ ≤ a ≤ b ≤ ∞. Consider now first t k-1 ≤ t < t k and t k -1 ≤ t < t k , k = k . Then E u,v t,t = t t k-1 t t k -1 ∂ 2 r,r ϕ(u, r; v, r )dr dr = t t k-1 t t k -1 (|v| α |u + v| α -|v| 2α )ϕ(u, r; v, r )dr dr.
Using |v| ≤ |u|+|u+v|, as well as distinguishing the cases |u+v| ≤ |u| and |u+v| > |u|, we have on the one hand for 0 < ε < 1

(1 ∧ |u| -1 )(1 ∧ |v| -1 ) ≤ (1 ∧ |u| -1 )|v| -2 (2|u| + 2|u + v| 1+ε |u| -ε ) ≤ 2|v| -2 (1 + (|u| -ε ∧ |u| -1-ε )|u + v| 1+ε ), (11)
and on the other hand

(1 ∧ |u| -1 )(1 ∧ |v| -1 )|u + v| α ≤ (1 ∧ |v| -1 )(|u + v| α-1 + (|u| -ε ∧ |u| -1-ε )|u + v| α+ε ) ≤ |v| -1 (|u + v| α-1 + (|u| -ε ∧ |u| -1-ε )|u + v| α+ε ). ( 12
) Note that |u + v| β e -|u+v| α r/2 r -β/α for β ≥ 0, R |v| γ e -|v| α (r -r) dv (r -r) -(γ+1)/α for γ > -1, as well as R e -|u+v| α r du r -1/α , R (|u| -ε ∧ |u| -1-ε )du 1.
Multiplying [START_REF] Engelbert | The Tanaka formula for symmetric stable processes with index α, 0< α[END_REF] by |v| 2α and ( 12) by |v| α , yields then in [START_REF] Comte | Super optimal rates for nonparametric density estimation via projection estimators[END_REF], as long as 1 + ε < α and using the upper bound

r ε/α ≤ T ε/α , |E t,t | t k t k-1 t k t k -1 ((r -r) -2+1/α (r -1/α + r -(1+ε)/α ) + (r -r) -1 (r -1 + r -1-ε/α ))dr dr. (1 ∨ T ε/α ) t k t k-1 t k t k -1 ((r -r) -2+1/α r -(1+ε)/α + (r -r) -1 r -1-ε/α )dr dr (1 ∨ T ε/α )(t -(1+ε)/α k-1 ∆ 1/α n + t -1-ε/α k-1 ∆ n log n).

This proves (i). Let now t

k-1 < t < t < t k . To compute E[g(X t k-1 ) 2 ], we use (9) to obtain E[g(X t k-1 -ε )g(X t k-1 +ε )
] and let ε → 0. Then, (10) still holds, but this time with

E u,v t,t = t t k-1 (∂ r ϕ(u, r; v, t ) -∂ r ϕ(u, r; v, t k-1 ))dr = t t k-1 ((|v| α -|u + v| α )ϕ(u, r; v, t ) + |u| α ϕ(u, r; v, t k-1 ))dr.
Similar as above, after multiplying [START_REF] Engelbert | The Tanaka formula for symmetric stable processes with index α, 0< α[END_REF] by |v| α , we have

R 2 (1 ∧ |u| -1 )(1 ∧ |v| -1 )|v| α ϕ(u, r; v, t )d(u, v) (t -r) -1+1/α (r -1/α + r -(1+ε)/α ).
Moreover, by symmetry in u, v and because ϕ(u, r, v, t k-1 ) = ϕ(v, t k-1 , u, r), the same upper bound follows with respect to |u| α ϕ(u, r; v, t k-1 ) when t -r, r are replaced by r -t k-1 , t k-1 . At last, using |u| ε |u + v| ε + |v| ε and arguing as after [START_REF] Ibragimov | Independent and Stationary Sequences of Random Variables[END_REF], we find that

R 2 (1 ∧ |u| -1 )(1 ∧ |v| -1 )|u + v| α ϕ(u, r; v, t )d(u, v) R 2 |u + v| α |v| ε (|u + v| ε + |v| ε )(|u| -ε ∧ |u| -1-ε )(|v| -ε ∧ |v| -1-ε )ϕ(u, r; v, t )d(u, v) r -1-ε/α (t -r) -ε/α + r -1 (t -r) -2ε/α .
Combining the last two displays, we find as in (i) from [START_REF] Comte | Super optimal rates for nonparametric density estimation via projection estimators[END_REF] for sufficiently small ε

|E t,t | ∆ 1/α n (t -1/α k-1 + t -(1+ε)/α k-1 ) + ∆ 1-ε/α n t -1-ε/α k-1 + ∆ 1-2ε/α n t -1 k-1 .

Proof of Theorem 4

Note that LT,n (y) = (2h n ) -1 ÔT,n ([y -h n , y + h n ]). By Theorem 2 for 1 < α ≤ 2 and sufficiently small ε > 0 we then get

LT,n (y) -L T (y) 2 L 2 (P) (2h n ) -1 O T ([y -h n , y + h n ]) -L T (y) 2 L 2 (P) + (1 ∨ T ε )h -2 n T 1-1/α-ε ∆ 1+1/α n . (13)
The occupation time formula (4) yields

O T ([y -h n , y + h n ]) = [y-hn,y+hn] L T (x)dx such that because of [y-hn,y+hn] dx = 2h n (2h n ) -1 O T ([y -h n , y + h n ]) -L T (y) L 2 (P) ≤ 1 2 1 -1 L T (y + h n x) -L T (y) L 2 (P) dx = T 1-1/α 2 1 -1 L 1 (T -1/α (y + h n x)) -L 1 (T -1/α y) L 2 (P) dx,
where we used the self-similarity of X, which also transfers to its local time, cf. Proposition 10.4.8 of [START_REF] Samorodnitsky | Stochastic Processes and Long Range Dependence[END_REF]. A typical moment bound for local times, see for example [16, Lemma 5.2], therefore implies that the last display is upper bounded up to a constant by

T 1-1/α (T -1/α h n ) (α-1)/2 = T 1/2-1/(2α) h (α-1)/2 n .
Using this in (13) yields the claim.

Proofs of optimal estimation results

We first present some results on conditional expectations.

Lemma 16. Suppose that y ∈ R and

Z d = X 1 is independent of X. (i) O T (y) -E[O T (y)|G n ] 2 L 2 (P) = ∆ 2 n n-1 k=0 E Var O 1 (k 1/α Z + ∆ -1/α n y) Z, X 1 . (ii) If 1 < α ≤ 2, then with C(α) = -(α -1)Γ(α) cos( πα 2 ) L T (y) -E[L T (y)| G n ] 2 L 2 (P) = ∆ 2-2/α n C(α) -2 n-1 k=0 E Var L 1 (k 1/α Z + ∆ -1/α n y) Z, X 1 . Proof. (i). Set g(x) = 1(x ≥ y), x ∈ R, such that O T (y) -E[O T (y)|G n ] = n k=1 t k t k-1 (g(X r ) -E[g(X r )|G n ])dr.
By independence and stationarity of the increments, the Markov property implies for

t k-1 ≤ r ≤ t k E [g(X r )| G n ] = E g (X r )| X t k-1 , X t k -X t k-1 = E g X r -X t k-1 + X t k-1 X t k-1 , X t k -X t k-1 (14) d
= E g ∆ 1/α n X (∆n) -1 (r-t k-1 ) + t 1/α k-1 Z Z, X 1 , concluding by X r -X t k-1 d = ∆ 1/α n X (∆n) -1 (r-t k-1 ) , X t k-1 d = t 1/α
k-1 Z due to self-similarity. In particular, the random variables

t k t k-1 (g(X r ) -E[g(X r )|G n ])dr being uncorrelated for different k, we obtain O T (y) -E[O T (y)|G n ] 2 L 2 (P) = n k=1 E Var t k t k-1 g(X r )dr X t k-1 , X t k -X t k-1 = ∆ 2 n n k=1 E Var 1 0 g(∆ 1/α n X r -t 1/α k-1 Z)dr Z, X 1 .
The result follows from g(∆

1/α n X r -t 1/α k-1 Z) = 1(X r ≥ (k -1) 1/α Z + ∆ -1/α n y). (ii).
Applying the Itô-Tanaka formulas of [START_REF] Protter | Stochastic differential equations[END_REF] for α = 2 and of [11, Theorem 2.1] for 1 < α < 2 we find

C(α)L T (y) = |X T -y| α-1 -|y| α-1 -M T (y)
for a square integrable martingale (M t (y)) t such that for all 0 < s < t, M t (y) -M s (y) is independent of σ(M u (y), u ≤ s). Indeed, for α = 2 we have M t (y) = t 0 sgn(X r -y)dX r and for 1 < α < 2

M t (y) = t 0 R |X u --y + x| α-1 -|X u --y| α-1 q(du, dx),
with q being the compensated Poisson random measure associated with X. As in (i),

E[ M k∆n (y) -M (k-1)∆n (y) G n ] = E[M k∆n (y) -M (k-1)∆n (y) X t k -1 , X t k -X t k-1 ] d = E[M ∆n (y -t 1/α k-1 Z) Z, X ∆n ],
and the random variables (M k∆n (y) -M (k-1)∆n (y) -E[M k∆n (y) -M (k-1)∆n (y)|G n ]) k are uncorrelated. Consequently,

C(α) 2 L T (y) -E[L T (y)| G n ] 2 L 2 (P) = M T (y) -E[M T (x)| G n ] 2 L 2 (P) = n k=1 E Var M ∆n (y -t 1/α k-1 Z) Z, X ∆n = n-1 k=0 E Var L ∆n (t 1/α k Z + y) Z, X ∆n ,
using the symmetry Z d = -Z in the last line. The result follows from self-similarity of X and Proposition 10.4.8 of [START_REF] Samorodnitsky | Stochastic Processes and Long Range Dependence[END_REF], which implies that ∆

1/α-1 n L ∆nt (∆ 1/α n a) is a version of the local time L t (a).
Proof of Theorem 5. Observe the equality in Lemma 16(i). Since the first summand in that sum is of order O(∆ 2 n ) and thus asymptotically negligible, we only have to study the sum for k ≥ 1, which equals

∆ 2 n n-1 k=1 R E Var O 1 k 1/α x + ∆ -1/α n y X 1 f α (x)dx. ( 15
)
Let first 1 < α ≤ 2. After a change of variables the last line equals

R E [Var ( O 1 (x)| X 1 )] ∆ 2+1/α n n-1 k=1 t -1/α k f α (∆ 1/α n t -1/α k x -t -1/α k y)dx.
Since f α is uniformly bounded, dominated convergence implies

∆ n n-1 k=1 t -1/α k f α (∆ 1/α n t -1/α k x -t -1/α k y) → T 0 t -1/α f α (t -1/α y)dt = T 0 f α,t (y)dt.
This equals E[L T (y)] by the occupation time formula (4), and the claim follows from using dominated convergence again. Let now 0 < α < 1 and suppose y = 0. Decompose the expected value in ( 15) as

E[Var(Y |X 1 )] = E[Y 2 -E[Y |X 1 ] 2 ] with Y = O 1 (k 1/α x) = 1 0 1(X r ≥ k 1/α x)dr. Multiplying out the square yields n-1 k=1 O 1 (k 1/α x) 2 = 1 0 1 0 n-1 k=1 1 k ≤ |X r ∧ X s | α x α , X r ∧ X s ≥ 0 drds = 1 0 1 0 |X r ∧ X s | α x α ∧ (n -1) 1(X r ∧ X s ≥ 0)drds.

Auxiliary lemmas for asymptotic results

In this section, we collect a number of technical properties for α-stable distributions. We begin by recalling a well-known asymptotic property in the case α ≤ 1. For a proof see [START_REF] Ibragimov | Independent and Stationary Sequences of Random Variables[END_REF]Theorem 2.4.2], or [23, page 246] for the value of the limit.

Lemma 17. If 0 < α ≤ 1, then h α (x) = x 1+α f α (x), x ≥ 0, is a non-negative nondecreasing function satisfying

lim x→+∞ h α (x) = α π sin( πα 2 )Γ(α) =: h α (∞). ( 18
)
The next two lemmas are used in Theorems 5 and 3.

Lemma 18. For 0 < α ≤ 1 let ρ r,s denote the Lebesgue density of X r ∧ X s for 0 < r, s ≤ 1, f α,s|X 1 is the conditional Lebesgue density of X s for 0 < s < 1, conditional on X 1 and set ρ r,s,X

1 (z) = 2f α,r|X 1 (z) ∞ z f α,s|X 1 (z)dz, z ≥ 0. Then, as z → ∞: (i) z 1+α ρ r,s (z) → (r ∧ s)h α (∞), (ii) z 1+α E[ρ r,s,X 1 (z)] → rsh α (∞). (iii) π 2 (z/r) 2 log(z/r) ∞ 0 xf 1 (x)ρ r,s (xz)dx → 1 r∧s if α = 1. (iv) π 2 (z/r) 2 log(z/r) ∞ 0 xf 1 (x)E[ρ r,s,X 1 (xz)]dx → r∨s r∧s if α = 1. Proof. By independence of increments it is not difficult to see for 0 < r < s that ρ r,s (u) = 1 2 f α,r (u) + ∞ 0 f α,r (u + v)f α,s-r (v)dv. ( 19
)
Monotone convergence and Lemma 17 yield (i). On the other hand, again by independence of increments, the conditional density for 0 < r < 1 is given by

f α,r|X 1 =x (z) = f α,r (z)f α,1-r (x -z) f α,1 (x) , recalling that f α,1 = f α . Denote Ψ r (z, x) = ∞ z f α,r|X 1 =z+x (z)dz.
As above, monotone convergence and Lemma 17 imply then as z → ∞

Ψ r (z, x) = 1 (z + x) 1+α f α,1 (z + x) ∞ 0 (z + x) 1+α f α,r (z + z)f α,1-r (x -z)dz → 1 h α (∞) ∞ 0 rh α (∞)f α,1-r (x -z)dz = r ∞ -x f α,1-r (z)dz,
using symmetry of f α,1-r in the last line. Since also trivially Ψ(z, x) ≤ Ψ(-∞, x) = 1, we find from this by dominated convergence

z 1+α E[ρ r,s,X 1 (z)] = 2z 1+α R f α,r (z)f α,1-r (x -z) f α,1 (x) ∞ z f α,s|X 1 =x (z)dzf α,1 (x)dx = 2z 1+α f α,r (z) R f α,1-r (x)Ψ s (z, x)dx (20) → 2rsh α (∞) R f α,1-r (x) ∞ -x f α,1-s (z)dzdx = rsh α (∞),
again using symmetry of the densities. This shows (ii). For (iii) let again 0 < r < s.

From ( 19) we find that

∞ 0 xf 1 (x)ρ r,s (xz)dx = 1 2r g 1 (0, z r ) + 1 r ∞ 0 g 1 ( v r , z r )f α,s-r (v)dv, where g 1 (a, b) = ∞ 0 xf 1 (x)f 1 (a + bx)dx. Lemma 21 implies for z → ∞ π 2 (z/r) 2 log(z/r) ∞ 0 xf 1 (x)ρ r,s (xz)dx → 1 2r + 1 r ∞ 0 f 1,s-r (v)dv = 1 r .
At last, starting from (20), we have

∞ 0 xf 1 (x)E[ρ r,s,X 1 (xz)]dx = 2 ∞ 0 xf 1 (x)f α,r (xz) R f α,1-r (x)Ψ s (xz, x)dxdx.
For α = 1, the density is f 1 (x) = 1 π(1+x 2 ) and so (z/r) 2 log(z/r) xf 1 (x)f α,r (xz) is clearly integrable uniformly in z ≥ 0. Since Ψ s (xz, z) ≤ 1 by (ii) and Ψ s (xz, x) → s ∞ -x f α,1-s (z)dz, the last display equals, as z → ∞,

2s ∞ 0 xf 1 (x)f α,r (xz) R f α,1-r (x) ∞ -x f α,1-s (z)dzdx + o log(z/r) (z/r) 2 = s r g 1 (0, z r ) + o log(z/r) (z/r) 2 .
We conclude with Lemma 21.

Lemma 19. Let f D denote the Lebesgue density of D = (1 + | Z/Z| α ) -1 . (i) If 1 < α ≤ 2, then sup 0<x<1/2 |x 1-1/α f D (x)| < ∞ and x 1-1/α f D (x) → 2 πα 2 Γ( 1 α )E[|Z|] as x → 0. (ii) If 0 < α < 1, then sup 0<x<1/2 f D (x) ≤ 4 and f D (x) → 2 π sin( πα 2 )Γ(α)E[|Z| -α ] as x → 0. (iii) If α = 1, then f D (x) = 4 π 2 (1 -2x) -1 log(x -1 -1).
Proof. For all 0 < α ≤ 2 denote by g α (x) = 4 ∞ 0 yf α (y)f α (xy)dy, x > 0, the density of | Z/Z|, such that

f D (x) = 1 α x -1-1/α (1 -x) -1+1/α g α ((x -1 -1) 1/α ), 0 < x < 1. ( 21
) (i). For 1 < α ≤ 2, E[|Z|] is finite and therefore g α (x) ≤ 2 sup y>0 f α (y)E[|Z|].
Together with the property g α (x) = x -2 g α (x -1 ) for x = 0, we get from (21) for 0 < x < 1/2 that

|x 1-1/α f D (x)| x -2/α (1 -x) -1+1/α (x -1 -1) -2/α = (1 -x) -1-1/α 1.
The claimed convergence follows from g α (x) → 2f α (0)E[|Z|] as x → 0 and from f α (0) = (απ) -1 Γ(1/α).

(ii). Observe first that x(1-x/n)(1-2x/n) dx = 1 12 .

x 1+α g α (x) = 4
Proof. (i). Using the relationship x -1 (1 -x/n) -1 -1 = n -1 (1 -x/n) -1 ≤ 2n -1 for 0 < x < n/2 and ψ(x) ≤ 1, it is enough to study the limit of 1 log n n/2 0 ψ(x)

x dx. Moreover, since we have for n = 2k + 1 odd the bound x -x 2 j + x dx.

Since j -1 -(j + x) -1 ≤ j -2 for 0 < x < 1 is summable in j, we conclude that lim (ii). Note that

1 x(1 -x/n)(1 -2x/n) = 1 x + 2 n(1 -2x/n) + 1 n(1 -x/n)(1 -2x/n) .
For 0 < x < n/2 the last term is bounded by the second one. Since where we used log((x/n) -1 -1) = log n + log(1 -x/n) -log x in the last line. We show below that the second term equals -1/12. Together with the result from (i) we get (ii). In order to find the limit for the second term, note that as in (i) it is enough to consider n = 2k even. As above, we have Lemma 21. Let α = 1 and define

g 1 (a, b) = ∞ 0 xf 1 (x)f 1 (a + bx)dx = 1 π 2 ∞ 0 x (1 + x 2 )(1 + (a + bx) 2 )
dx.

Then for every a ≥ 0 and b > 0, we get the following equality

g 1 (a, b) = 1 π 2 ((1 + a 2 -b 2 ) 2 + 4a 2 b 2 ) 1 + a 2 -b 2 2 log( 1 + a 2 b 2 ) + πab -a(1 + a 2 -b 2 )( π 2 -arctan(a)) . (22)
In particular,

g 1 (0, b) = log b π 2 (b 2 -1)
, lim

3 / 4 n

 34 for O T (A) and ∆ 1/4

Theorem 4 .

 4 Let 1 < α ≤ 2 and let y ∈ R. Then for any sufficiently small ε > 0

Figure 1 :

 1 Figure 1: Numerical illustration of the function α → Cα .

Remark 8 .Remark 9 .

 89 For α = 2 Theorems 5 and 7 can be obtained from [13, Theorem 3]. Let us discuss some further properties of ÔT,n (A) and E[O T (A)|G n ].

3. 1 . 1

 11 Proof of Theorem 3Let us decompose V T,n := O T (0) -ÔT,n (0) 2 L 2 (P) as

∞ 0 f

 0 α (y) y α h α (xy)dy is non-decreasing in x > 0 and converges to 2h α (∞)E[|Z| -α ] as x → ∞ by the monotone convergence theorem and Lemma 17. This yields with[START_REF] Protter | Stochastic differential equations[END_REF] for 0 < x < 1/2 f D (x) x -1-1/α (1 -x) -1+1/α (x -1 -1) -1-1/α = (1 -x) -2 < 4,and the claimed convergence of f D (x) as x → 0.(iii). For α = 1, the equality is a straightforward consequence of Lemma 21 and the equality f D (x) = 4x -2 g 1 (0, x -1 -1).Lemma 20. Recall the function ψ(x) = (x -x ) -(x -x ) 2 , x ≥ 0, from Theorem 3) log((x/n) -1 -1)

  dx ≤ 1/k 1/n, we only have to consider n = 2k even. Denoting by [x] = x -x the fractional part of x ∈ R, we have

1 0

 1 (x -x 2 )dx = 1 6 .

  x 2 ) log(j + x) j + x dx.Noting that | log(j+x) j+x -log j j | ≤ j -2 for 0 < x < 1 is summable, the claimed limit is obtained

b→+∞ π 2 b 2 log b g 1

 1 (a, b) = 1, and g 1 (a, b) ≤ g 1 (0, b) = log b π 2 (b 2 -1) .

Writing similarly E[O 1 (k 1/α x)|X 1 ] = 1 0 R 1(z ≥ k 1/α x)f α,r|X 1 (z)dzdr with the conditional density f α,s|X 1 from Lemma 18, using symmetry and taking expectations allows to rewrite [START_REF] Khoshnevisan | Local times of additive Lévy processes[END_REF] as

with h(u) =

)drds for ρ r,s , ρ r,s,X 1 from Lemma 18. In order to conclude, consider the decomposition

By Lemma 18(i,ii), h(u) → hα(∞) 12 , u → ∞ and sup u≥0 |h(u)| < ∞, and so the expression in the last display converges uniformly in x ≥ 0, when divided by log n, to h α (∞)/(12α). The result is obtained from dominated convergence and noting 2

At last, let α = 1 and suppose again y = 0. As for 0 < α < 1 it suffices to study [START_REF] Marcus | Lp moduli of continuity of Gaussian processes and local times of symmetric Lévy processes[END_REF], which we rewrite as

Lemma 18(iii,iv) yield h(u)

, the claim follows from dominated convergence.

Proof of Theorem 7. According to Lemma 16(ii) and changing variables we have

By a typical moment bound for local times, cf. [16, Lemma 5.2], and the same Riemann sum convergence as after [START_REF] Khoshnevisan | Local times of additive Lévy processes[END_REF], we conclude by dominated convergence

Proof. We only compute

dx, all the other properties being easily deduced from [START_REF] Samorodnitsky | Stochastic Processes and Long Range Dependence[END_REF]. Writing K -1 a,b = (1 + a 2 -b 2 ) 2 + 4a 2 b 2 , we shall use the decomposition which yields [START_REF] Samorodnitsky | Stochastic Processes and Long Range Dependence[END_REF].


Thomas Schmitt
email: thomas.schmitt@centralesupelec.fr

Milan Pelletier

Sébastien Ducruix

Multi-fluid models for two-phase and transcritical flows Application to rocket engine configurations with the AVBP solver

Keywords: Two-phase flow, Supercritical flow, Non-Ideal thermodynamics, Large-Eddy Simulation

In various industrial combustion devices, such as liquid rocket engines or Diesel engines, the operating point varies over a wide range of pressure. These pressure variations can lead to a change of thermodynamic regime, switching from two-phase injection to transcritical injection. This change may modify the topology of the flow and the mixing, thereby impacting the flame dynamics. In the present contribution, a framework that is able to simulate compressible flows under both subcritical and supercritical states within the same solver is presented. This is achieved using a multifluid approach coupled with a cubic equation of state (EoS). Liquid/vapor equilibrium is computed within the binodal region to ensure the convexity of the EoS. Examples of coaxial flame simulations are provided, showing good agreements with experimental visualizations.

Introduction

The jet formed during liquid injection is strongly dependent on the operating temperature and pressure conditions. When pressure and temperature are low enough with respect to the critical point, the fluid undergoes a classical break-up process. The interfaces correspond to discontinuities between the liquid and gas phases. As pressure or temperature are increased to supercritical conditions, surface tension vanishes and the phase discontinuity that is observed at lower pressure is no longer present. Instead, the jet evolves in the presence of a continuous interface between the high-density stream and the surrounding gases. Thermodynamic properties feature strong -but continuous -variations across a diffuse interface. Under such conditions, the fluid injection regime is often referred to as transcritical and the jet mixing is controlled by turbulence and is analogous to that of a variable density jet [START_REF] Candel | Structure and dynamics of cryogenic flames at supercritical pressure[END_REF].

Supercritical and transcritical injection modeling has been deeply investigated during the last decades [START_REF] Oefelein | Mixing and combustion of cryogenic oxygen-hydrogen shear-coaxial jet flames at supercritical pressure[END_REF][START_REF] Zong | Cryogenic fluid jets and mixing layers in transcritical and supercritical environments[END_REF]. One important ingredient of such models is the appropriate description of the real-gas thermodynamics that drives the fluid behavior in such high pressure / low temperature situations. Thermodynamics generally rely on the use of a cubic equation of state, such as the Soave Redlich Kwong EoS [START_REF] Soave | Equilibrium constants from a modified Redlich-Kwong equation of state[END_REF]. In the context of Large-Eddy Simulations, good results have been obtained for the simulation of cryogenic coaxial flames, with satisfactory agreements in terms of flame topology between simulations and experiments [START_REF] Zips | Efficient thermo-chemistry tabulation for non-premixed combustion at highpressure conditions[END_REF][START_REF] Schmitt | Large-eddy simulations of the Mascotte test cases operating at supercritical pressure[END_REF].

For compressible cases, the strategies encountered in the literature for the modeling of separate two-phase flows mostly focus on diffuse interface methods and more specifically on multi-fluid methods [START_REF] Baer | A two-phase mixture theory for the deflagration-to-detonation transition (ddt) in reactive granular materials[END_REF][START_REF] Saurel | A multiphase Godunov method for compressible multifluid and multiphase flows[END_REF]. These methods seem well-adapted by construction to handle the interface appearance and disappearance that may occur in the targeted applications. They rely on an ensemble averaging of the phases properties to formulate sets of equations that rule the two-phase flow evolution, which are hyperbolic provided that convex thermodynamic closures are used [START_REF] Saurel | A multiphase Godunov method for compressible multifluid and multiphase flows[END_REF]. Under the assumption of pressure, temperature and chemical potentials equilibria between phases, simplified multifluid models can be derived, convenient to treat numerically [START_REF] Martelot | Towards the direct numerical simulation of nucleate boiling flows[END_REF]. Such models have been recently coupled with cubic EoS and a multicomponent two-phase equilibrium solver in a classic finite-volume framework [START_REF] Matheis | Multi-component vapor-liquid equilibrium model for LES of high-pressure fuel injection and application to ECN spray A[END_REF][START_REF] Traxinger | Single-phase instability in non-premixed flames under liquid rocket engine relevant conditions[END_REF], showing very good results with available experimental data in the context of turbulent multi-component jet injection.

In the present contribution, a multi-fluid approach is combined with a cubic equation of state in the finite-element LES solver AVBP [START_REF] Schönfeld | Steady and unsteady flows simulations using the hybrid flow solver avbp[END_REF][START_REF] Schmitt | Large-eddy simulations of the Mascotte test cases operating at supercritical pressure[END_REF][START_REF] Pelletier | A multifluid taylor-galerkin methodology for the simulation of compressible multicomponent separate two-phase flows from subcritical to supercritical states[END_REF] to simulate coaxial cryogenic flames operating at sub-and supercritical pressure. The governing equations and models are detailed in Sec. 2 and the thermodynamic closure is given in Sec. 3. Finally, examples of coaxial flame simulations are provided in Sec. 4.

Governing equations

Both 3-and 4-equation models have been integrated in the AVBP solver (see [START_REF] Pelletier | A multifluid taylor-galerkin methodology for the simulation of compressible multicomponent separate two-phase flows from subcritical to supercritical states[END_REF] for details). The three equation model, used in the simulations presented below, is detailed in this section. This model being similar to Euler equations used in gaseous and supercritical flows, similar closure are used here [START_REF] Schmitt | Large-eddy simulations of the Mascotte test cases operating at supercritical pressure[END_REF]. In the context of Large-Eddy Simulation, the corresponding Favre-filtered, fully compressible Navier-Stokes equations are given by:

∂ρ Y k ∂t + ∂ρ Y k u j ∂x j = - ∂J k,j ∂x j - ∂J t k,j ∂x j + ωk ( 1
)
∂ρ u i ∂t + ∂ρ u i u j ∂x j = - ∂p ∂x i + ∂τ i,j ∂x j + ∂τ t i,j ∂x j (2) ∂ρ E ∂t + ∂ρ u j E ∂x j = - ∂p u j ∂x j + ∂ u i τ i,j ∂x j - ∂q j ∂x j - ∂q t j ∂x j + ωT (3)
where φ and φ denote spatial and mass-weighted (Favre) spatially filtered quantities. p is the pressure, T the temperature, ρ the density, Y k is the mass fraction of the species k, u i represents the velocity vector components, x i the spatial coordinates, t is the time, E the total sensible energy, τ t i,j the sub-grid scale (SGS) stress tensor, q t j the SGS energy fluxes, J t k,j the SGS species fluxes, ωk the species reaction rate and ωT the heat release rate. The species J k and heat fluxes q use classical gradient approaches. The fluid viscosity and the heat diffusion coefficient are calculated following Chung et al. method [START_REF] Chung | Generalized multiparameter correlation for nonpolar and polar fluid transport properties[END_REF] and mass diffusion coefficients are deduced from heat diffusivity by assuming a unity Lewis number (Le=1). Soret and Dufour effects are neglected. The sub-grid scale (SGS) energy and species fluxes are modeled using the gradient transport assumption with turbulent Prandtl and Schmidt numbers both set to 0.7. The SGS turbulent viscosity is modeled with the wall-adapting large eddy (WALE) model [START_REF] Nicoud | Subgrid-scale stress modelling based on the square of the velocity gradient tensor[END_REF]. In the present work, combustion is modeled assuming infinitely fast reactions and pure diffusion regime operation [START_REF] Schmitt | Large-eddy simulations of the Mascotte test cases operating at supercritical pressure[END_REF].

Thermodynamic closure

Mixture properties are computed using the Soave-Redlich-Kwong (SRK) EoS [START_REF] Soave | Equilibrium constants from a modified Redlich-Kwong equation of state[END_REF]:

p(ρ, T, Y ) = ρrT 1 -b m ρ - a m (T )ρ 2 1 -b 2 m ρ 2 (4)
with the mixture covolume b m and attractive coefficient a m computed following the van der Waals mixing laws [START_REF] Poling | The properties of gases and liquids[END_REF]. In the subcritical domain, single-phase states can become unstable, leading to phase separation. The instability can be mechanical or chemical in the case of multicomponent mixtures, corresponding respectively to a loss of thermodynamic convexity along the pressure direction or along the chemical composition directions. Convexity can be restored by computing an equilibrium between a liquid and a vapor phase. In the present contribution, an approximation of the exact multi-component equilibrium is used assuming that, for each species, the liquid and vapor phase mass fractions are equal. Such an approximation is reasonable as long as phase separation is not dominated by a chemical instability. Practical methods for its calculation as well as more elaborated equilibrium closures are detailed in [START_REF] Pelletier | A multifluid taylor-galerkin methodology for the simulation of compressible multicomponent separate two-phase flows from subcritical to supercritical states[END_REF][START_REF] Pelletier | Diffuse interface models and adapted numerical schemes for the simulation of subcritical to supercritical flows[END_REF].

4 Application to the Mascotte test-cases

Simulated cases and numerical setup

The Mascotte experimental configuration of ONERA is considered here [START_REF] Habiballah | Experimental studies of high-pressure cryogenic flames on the Mascotte facility[END_REF][START_REF] Candel | Structure and dynamics of cryogenic flames at supercritical pressure[END_REF]. The present simulations reproduce cases A10 and C10 [START_REF] Kendrick | Effects of a recess on cryogenic flame stabilization[END_REF]. Simulation of cases operating at 60 bar (i.e. under transcritical conditions) can be found in [START_REF] Schmitt | Large-eddy simulations of the Mascotte test cases operating at supercritical pressure[END_REF]. A single coaxial injector produces a liquid oxygen stream at low velocity (less than 10 m/s), surrounded by a high-velocity gaseous hydrogen stream (more than 100 m/s), in a chamber at 10 bar. Liquid oxygen is injected at 80 K, well below its critical value of 155 K, while gaseous hydrogen is injected at 289 K. Under such conditions, the density of oxygen (ρ O2 = 1190 kg.m -3 ) is much larger than that of hydrogen (ρ H2 = 0.85 kg.m -3 ). Injection conditions are summarized in Tab. 4.1 The compressible unstructured solver AVBP [START_REF] Schönfeld | Steady and unsteady flows simulations using the hybrid flow solver avbp[END_REF] is used for this study. Its Taylor-Galerkin weighted residual central distribution scheme, called TTGC, is third-order in time and space [START_REF] Colin | Development of high-order Taylor-Galerkin schemes for LES[END_REF]. The Jacobian matrices of non-viscous fluxes used by the scheme and the characteristic boundary conditions are written to be fully consistent with the non-ideal thermodynamics [START_REF] Pelletier | A multifluid taylor-galerkin methodology for the simulation of compressible multicomponent separate two-phase flows from subcritical to supercritical states[END_REF].

Preliminary results

On going simulations results are shown in this section. A longitudinal slice of instantaneous temperature field for case A10 is shown in Fig. 1. A fully turbulent diffusion flame emanates from the coaxial injector. The white iso-contour shows the regions of liquid/vapor coexistence, which is essentially located after the liquid jet break-up. Simulations are qualitatively compared with experimental results in Fig. 2, where mean OH mass fraction field is compared with OH* emission exprimental pictures from Kendrick et al. [START_REF] Kendrick | Effects of a recess on cryogenic flame stabilization[END_REF]. For both cases, the initial open angle is well reproduced and the flame shape is qualitatively retrieved. However, flames seem to be slightly longer in the simulations compared with the experiment. AVBP solver and permits the simulation of subcritical to supercritical cryogenic flames. Results are in satisfactory agreement with experimental visualizations.

Fig. 1

 1 Fig.1Longittudinal slice of instantaneous temperature for case A10. Blue: 80 K, Red: 3400 K. The white isocontour represents liquid/vapor coexistence regions where the mixture is thermodynamically unstable.

Fig. 2

 2 Fig. 2 Comparison with experimental visualizations. Comparison between Abel's transform of experimental OH * mean emission for case A10 and C10 [6] (bottom part of the pictures) and a longitudinal cut of mean OH mass fraction (Blue: 0.01, red: 0.05) from LES (top part of the pictures). Dashed lines show the position of maximum OH* emission from experiments. Note that the experimental visualization for case A10 is not perfectly symmetrical, comparison is thus qualitative.

Table 1

 1 Injection conditions for the simulated cases. ṁ is the mass flow rate, Tr=T/Tc is the reduced temperature and Pr=p/pc is the reduced pressure, Tc and Pc being the critical temperature and pressure, respectively.

	Case Pressure [bar]	ṁH 2 [g/s]	ṁO 2 [g/s] Tr O 2	Pr O 2
	A10	10.0	23.7	50	0.52	0.20
	C10	10.0	15.8	50	0.52	0.20

Acknowledgements This work was granted access to HPC resources made available by GENCI (Grand Equipement National de Calcul Intensif) under the allocation A0082B06176. A part of this work was performed using HPC resources from the mesocentre computing center of Ecole CentraleSupélec and Ecole Normale Supérieure Paris-Saclay supported by CNRS and Région Ile-de-France.