Goodness-of-t test for multistable Lévy processes

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Goodness-of-t test for multistable Lévy processes

Ronan Le Guével

To cite this version:

Ronan Le Guével. Goodness-of-t test for multistable Lévy processes. Communications in Statistics - Theory and Methods, Taylor & Francis, In press, pp.1-31. �10.1080/03610926.2019.1653922�. �hal- 02275584�

Goodness-of-t test for multistable Lévy processes

R. Le Guével

Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France [email protected]

Abstract

The multistable processes are extensions of stable processes, where the index of stability is replaced by a function ranging in (0,2). The aim of this article is to build a statistical test which is able to detect a multistable behavior of a process belonging to the class of the multistable Lévy processes. The properties are stated for a discrete observation scheme of one trajectory.

Keywords: Multistable process, Lévy process, nonparametric test, energy test.

1 Introduction

Most of time, homogeneous processes are used for real datasets as rst intention models. If one deals with processes with jumps, one can use a Lévy process (or a transformation of it) as a basic model. Various generalisations of this process have been introduced these last years in order to overcome the lack of exilibility of these models. We can mention for instance the introduction of the Multifractional Brownian Motion (Peltier and Lévy-Véhel 1995; Benassi, Jaard, and Roux 1997; Ayache and Lévy-Véhel 2000; Herbin 2006), or the Linear Multifractional Stable Motion (Stoev and Taqqu 2004, 2005) as models where the regularity of the observation is tuned by a function H, which is playing the role of the local Hurst index. For these models, one statistical challenge is to build a test in order to decide if the regularity H is constant or not along the trajectory. One can see for instance (Biermé and Richard 2010; Richard and Vu 2019) where the authors deal with anisotropic multifractional Brownian elds. For processes with jumps, for the same reason it is interesting to deal with stochastic processes whose local Hölder exponent changes in a controlled manner, it is convenient to consider models where the jump intensity is allowed to vary in time.

Recall that a process is called α-stable (0< α≤2)if all its nite-dimensional distributions areα-stable (see Samorodnitsky and Taqqu 1994). A generalisation of these α-stable processes has been introduced in Falconer and Lévy-Véhel (2008) in order to provide good models for data containing a distribution of discontinuities which is varying. They are useful models for nancial records, EEG or natural terrains for instance. It is a practical way to deal with non-stationarities observed in various real-data phenomena, since a multistable process X is tangent, at each timeu, to a stable process Z_u in the following sense (Falconer 2002, 2003):

r→0lim

X(u+rt)−X(u)

r^h(u) =Z_u(t) (1)

for a suitable h(u)(the limit (1) is taken either in nite dimensional distributions or, when X has a version with càdlàg paths, in distribution - one then speaks of strong localisability).

Some properties of these processes has already been studied. We may mention for instance the behaviour of their distribution (Ayache 2013), some regularity properties (Falconer and Liu 2012; Biermé and Lacaux 2013; Le Guével and Lévy-Véhel 2013; Le Guével 2018) or some semi-martingale representations (Le Guével, Lévy-Véhel, and Liu 2012). The multistable subordinator provides also good models for processes in heterogeneous environments. We may nd some limit theorems for the subordinator and the multifractional Poisson processes in Molchanov and Ralchenko (2015). The study of its generator and some related dierential equations for multistable Markov processes are also major subjects (see e.g. Beghin and Ricciuti 2018; Orsingher, Ricciuti, and Toaldo 2016; Ricciuti and Toaldo 2017).

The remainder of this paper is organized as follows: the next section is devoted to the denitions of two versions of the Lévy multistable process, using their Ferguson-Klass-LePage representation. In section 3, we introduce general notations. We specify some estimation tools and we state some auxiliary properties about these tools. Section 4 presents the problem of testing the multistability of a Lévy multistable model. We give in this section the main results of convergence of our testing procedure. Section 5 is dedicated to technical lemmas, useful for the proofs of the main theorems. Finally, we have gathered all the proofs in section 6.

2 Multistable processes

For α ∈ (0,2], recall that the stochastic integral I(f) := R

f(x)M(dx) of a real function f with respect to M exists if, for instance M is a symmetric α-stable random measure on R, with the Lebesgue measure as the control measure, and if f is measurable and satises R

R|f(x)|^αdx < +∞ (see Samorodnitsky and Taqqu 1994). Many symmetric stable processes {Y(t), t∈R}admit the stochastic integral representation

Y(t) = Z

f_t(x)M(dx).

Write Sα(σ, β, µ) for the α-stable distribution with scale parameter σ, skewness β and shift parameter µ. The marginal distribution of Y is therefore Y(t) ∼ S_α(σ_ft,0,0) where σ_ft =

R

R|f_t(x)|^αdx1/α

.

We will restrict our attention to the standard symmetric α-stable Lévy process on the interval[0,1], which may be dened as

L_α(t) :=

Z

R

1_[0,t](x)M(dx), t∈[0,1].

Since L_α(t) ∼ S_α(t^1/α,0,0), the logarithm of the characteristic function of L_α(t) is given by logE^α[e^iθLα(t)] =−t|θ|^α.

A well-known representation of the stable Lévy process is its Ferguson-Klass-LePage series representation, based on the following sequences:

• (Γ_i)i≥1 a sequence of arrival times of a Poisson process with unit arrival rate,

• (V_i)i≥1 a sequence of i.i.d. random variables with uniform distribution on [0,1], indepen- dent of (Γ_i)_i≥1,

• (γ_i)i≥1 a sequence of i.i.d. random variables with distribution P(γ_i = 1) =P(γ_i =−1) = 1/2, independent of (Γi)i≥1 and (Vi)i≥1.

Consequently, the stable Lévy process{Lα(t), t∈[0,1]} admits the series representation:

L_α(t) =

∞

X

i=1

γ_iC_αΓ^−1/α_i 1_[0,t](V_i)

whereC_α = R∞

0 x^−αsinx dx^−1/α

. For more details about Ferguson-Klass-LePage representa- tions, we refer the reader to Ferguson and Klass (1972), Rosinski (1990) or Samorodnitsky and Taqqu (1994). The stable Lévy motion is therefore a càdlàg process, jumping at time V_i with a jump of sizeC_αΓ^−1/α_i , where the stability index α may be seen as a parameter regulating the size of the jumps.

The multistable processes are more exible models because they allow us to consider a non constant index of stability α. The size of the jumps will be governed by a function α(t) evolving with time. The rst way to dene such a process is to use the Ferguson-Klass-LePage representation of the stable processes, as in Le Guével and Lévy-Véhel (2012), replacingα by a functionα: [0,1]→(0,2). The rst denition of the multistable Lévy motion is then

L₁(t) =

∞

X

i=1

γ_iC_α(t)Γ^−1/α(t)_i 1_[0,t](V_i).

Since we have replaced α by α(t) ∈ (0,2), for each t ∈ [0,1], Y(t) is a symmetric α(t)-stable random variable S_α(t)(t^1/α(t),0,0)and logEα[e^iθL1(t)] =−t|θ|^α(t).

The second denition, developed by Falconer and Liu (2012), is inherited from the de- nition of multistable random measures Mα(x), where we have replaced again α by a function α(t). They dened the stochastic integral of f with respect to a multistable random measure providing all its nite dimensional distributions. The Ferguson-Klass-LePage representation of the multistable Lévy motion resulting from this denition is

L₂(t) =

∞

X

i=1

γ_iC_α(Vi)Γ^−1/α(V_i ⁱ⁾1_[0,t](V_i), which satiseslogEα[e^iθL2(t)] =−Rt

0 |θ|^α(x)dx.It follows that the two multistable processes have quite dierent properties. In particular, L2 is a Markov process with independent increments, whileL₁ has none of these two properties.

We already know that the two processes L₁ and L₂ are linked by the following formula (Le Guével, Lévy-Véhel, and Liu 2012, Theorem 8):

L₁(t) =W(t) +L₂(t), whereW(t) =

t

R

0 +∞

P

i=1

γ_iK_i(u)1_[0,u[(V_i)du and K_i(u) = ^d

Cα(s)Γ^−1/α(s)_i

ds (u).

In this paper, we will consider the case of L1 as well as the case of L2.

3 Notations

For the weak convergences, we will use the uniform metric, based on the Kolmogorov distance, denoted byκ. Recall that κ is dened on the set of all probability measures on R by

κ(µ1, µ2) = sup

x∈R

|F_µ1(x)−Fµ2(x)|

whereF_µ is the cumulative distribution function with respect toµ. From now on, we make the assumption thatα is a C¹ function which is ranging in [α∗, α^∗], a subset of (1,2).

Given any a and b in (0,1) and any M > 0, we consider the subsets of continuously dierentiable functions

Θ₀ ={α: [a, b]7→[α∗, α^∗] | ∀(t₁, t₂)∈[a, b]², α(t₁) =α(t₂)}

and

Θ₁ ={α: [a, b]7→[α∗, α^∗] | ∃(t₁, t₂)∈[a, b]², α(t₁)6=α(t₂) and kα⁰k∞≤M}.

If α is not continuously dierentiable, some results will no longer be valid, because we can not apply the mean value theorem anymore. The proofs may be adapted if α is Hölder continuous instead of Lipschitz. On the other side, the interesting case of change point detection problem with a piecewise constant function α is rather more challenging. This would be an interesting further development that would require new methods.

For simplicity of notation, we will write α instead of α(t) if α ∈ Θ0. If Z is a standard symmetricα-stable random variable (Z ∼S_α(1,0,0)), we know that ifα∈(1,2),

µ(α) :=Eα[log|Z|] = Γ⁰(1)(1− 1

α) (2)

and

var_α(log|Z|) = π² 6α² +π²

12. (3)

Therefore we introduce the two functionsv(x) = _6x^π22 +^π₁₂² andσ²(x) = (v(x)∨v(2))∧v(1), and we will denote

h(t) := 1

α(t). (4)

We deneX_k,N to be the incrementX(^k+1_N )−X(_N^k)of a processX,N ∈N\{0}, k = 1, ..., N and we denotet_k = _N^k. We also introduce the process

U_k,N(t) =

∞

X

i=1

γ_iC_α(tk)Γ^−1/α(t_i ^k)1_[0,t](V_i) (5) and its increments

U_k,N :=U_k,N(t_k+1)−U_k,N(t_k) =

∞

X

i=1

γ_iC_α(tk)Γ^−1/α(t_i ^k)1_(tk,tk+1](V_i). (6)

Note that for k 6= j, since (t_k, t_k+1] ∩ (t_j, t_j+1] = ∅, U_k,N is independent of U_j,N (see Le Guével and Lévy-Véhel (2012, Proposition 6.1) for the expression of the characteristic function of(U_k,N, U_j,N)). SinceU_k,N(t)is a standard symmetricα(t_k)-stable Lévy process, we know that N^1/α(tk)U_k,N ∼S_α(tk)(1,0,0)and

var_α(log|U_k,N|) = var_α(log|N^1/α(tk)U_k,N|) =σ²(α(t_k)). (7) In the sequel, for estimation and testing problems,(n(N))N∈Nwill be a sequence taking even integer values such that n(N) ≤ N. Several rates of convergence will depend on the sequence ξ_N :=p

n(N)(Γ⁰(1)+logN).Fort∈[a, b]⊂(0,1), with the observations(X_k,N)_k, we introduce an estimator ofh(t) by

hˆ_N(t) = 1 n(N)

bN tc+^n(N)₂ −1

X

k=bN tc−^n(N)₂

Γ⁰(1)−log|X_k,N|

Γ⁰(1) + logN , (8)

where b.c is the oor function (see Le Guével (2013) for the properties of this estimator). We will also consider a slight modication using the process U (which will not be observed) with

ˆh^U_N(t) = 1 n(N)

bN tc+^n(N)₂ −1

X

k=bN tc−^n(N)₂

Γ⁰(1)−log|Uk,N|

Γ⁰(1) + logN , (9)

and we shall write ¯h(t) for its expectation Eα[ˆh^U_N(t)], that is

¯h(t) = 1 n(N)

bN tc+^n(N)₂ −1

X

k=bN tc−^n(N)₂

h(k

N). (10)

We also introduce a plug-in estimator ofσ²(α(t)) with ˆ

σ²(t) = σ²( 1

ˆh_N(t)). (11)

Our test is based on straightforward properties, listed below, of the estimator of h which are satised when the observed process is a standard stable Lévy motion, that is whenα is no longer varying (α ∈Θ₀):

• If α∈Θ₀, for all N ∈N^∗ and t₀ ∈[a, b], E^α[ˆh_N(t₀)] =h = 1

α and var_α[ˆh_N(t₀)] = σ²(α)

n(N)(Γ⁰(1) + logN)². (12) Furthermore, if lim

N→+∞n(N) = +∞ and lim

N→+∞

n(N) N = 0,

N→+∞lim

(Γ⁰(1) + logN)p n(N)

pσ²(α) (ˆh_N(t₀)−h)=^d N(0,1).

• There exists c₀ >0such that for all ε >0 and t₀ ∈[a, b],

sup

α∈Θ0

P^α(|ˆh_N(t₀)−h|>2ε)≤ c^n(N)₀

e^{εn(N)(Γ0(1)+logN)}, (13) whereby for every t₀ ∈[a, b]and α ∈Θ₀, if lim

N→+∞n(N) = +∞, lim

N→+∞

ˆh_N(t₀)^Pα=^a.s.h= 1 α.

Finally, in all the paper,K stands for an absolute positive constant, and K^∗ will be a positive constant which depends on Θ₀ or Θ₁, that isa, b, α∗, α^∗ and M.

4 Testing the multistability

We assume that the observation ofX =L₁ or X =L₂ consists of one realization of the points X(_N^k), N ∈N^∗,k = 1, ..., N. Our purpose is to solve the following testing problem

(H₀) :α ∈Θ₀ vs (H₁) :α∈Θ₁.

Set(t₀, t₁)∈(a, b)²,t₀ 6=t₁, and assume that(t₀−^n(N)_2N , t₀+^n(N)_2N )∩(t₁−^n(N_2N⁾, t₁+^n(N)_2N ) =∅ (this is always available for N large enough as soon as lim

N→+∞

n(N)

N = 0). Our test is based on the statistic

T_N = n(N)(Γ⁰(1) + logN)² ˆ

σ²(t₁)(b−a)

b

Z

a

|ˆh_N(t)−ˆh_N(t₀)|²dt. (14) According to the asymptotic distribution ofT_N under the null hypothesis, we obtain an asymp- totic consistent test with asymptotic sizeβ ∈(0,1) choosing the rejection region

Rc := [qβ,+∞)

where q_β is the 1−β quantile of the distribution 1 +χ²(1), and χ²(1) is the χ² distribution with one degree of freedom.

The following statements are our main results. The rst theorem exhibits the asymptotic distribution of the statistic T_N under the null hypothesis, providing a uniform bound for the size of the test.

Theorem 1 (Null hypothesis). Let P_N^α be the distribution of TN and P_1+χ² be the distribution of1 +χ²(1),where χ²(1) stands for theχ² distribution with one degree of freedom. There exists a constant K^∗ >0 such that for all N ≥1,

sup

α∈Θ0

κ(P_N^α, P_1+χ²)≤K^∗( 1

ξ_N + 1 pn(N) +

rn(N) N + 1

ξ_N + 1

pbN bc − bN ac −n(N)

!1/3

).

(15)

Thereby there exists a constant K^∗ > 0 such that for every xed level β ∈ (0,1) in the testing problem and every N ≥1,

sup

α∈Θ₀Pα(T_N ∈R_c)≤β+K^∗( 1

ξ_N + 1 pn(N)+

rn(N) N + 1

ξ_N + 1

pbN bc − bN ac −n(N)

!1/3

).

(16) Furthermore, if lim

N→+∞n(N) = +∞ and lim

N→+∞

n(N)

N = 0, then the following convergence in distribution occurs for all α∈Θ0 :

Nlim→+∞T_N = 1 +^d χ²(1). (17)

The next theorem deals with the convergence of the test statistic T_N under the alternative hypothesis. We obtain a consistent procedure under mild conditions onn(N).

Theorem 2 (Alternative hypothesis for L₁). Assume that the observed process is L₁ (i.e.

X=L₁). Dene

χ_N := 1 ξ_N



T_N − ξ_N² ˆ σ²(t₁)

1 b−a

b

Z

a

|h(t)−h(t₀)|²dt





and consider χ, a random variable normally distributed N(0, σ_χ²) with

σ_χ² = 4σ²(α(t₀)) σ⁴(α(t₁))(b−a)²

Z b a

(h(t)−h(t₀))dt ²

.

Let P_χN be the distribution of χ_N, P_χ be the distribution of χ, and let η ∈ (0,¹₂). There exists a positive constant K_η^∗ such that for all α ∈Θ₁ and N ≥1,

κ(P_χN, P_χ)≤K_η^∗( 1

pn(N)+ 1

√σχ

s ξ_N N^{(1−α∗1 )η2}

+ξ_Nn(N)

N + (n(N)

N )¹² + 1 ξN

). (18) Thereby if n(N) = N^γ with γ ∈(0,^α_4α^∗−1

∗ ), then for any α ∈Θ₁,

Nlim→+∞χN

=d χ (19)

and for a xed level β ∈(0,1) in the testing problem, there exists a constant K_γ^∗ >0 such that for all α∈Θ₁ and N ≥1,

Pα(T_N ∈R_c)≥Pα(χ≤ ξ_N 3(b−a)

Z b a

|h(t)−h(t₀)|²dt− q_β

ξ_N)−K_γ^∗

√logN N^γ/4



 1 b−a|

b

Z

a

(h(t)−h(t₀))dt|





−¹

2

. (20)

If h(t₀) = _b−a¹ Rb

ah(t)dt, lim

N→+∞χ_N ^P= 0^α where the convergence is in probability. In fact it appears in the proof of Theorem 2 that in that case the following convergence occurs

lim

N→+∞T_N − ξ_N² ˆ σ²(t1)

1 b−a

b

Z

a

|h(t)−h(t₀)|²dt=^d 1 b−a

Z b a

σ²(α(t))

σ²(α(t1))dt+ σ²(α(t₀)) σ²(α(t1))χ²(1), and we may obtain a similar bound for the Kolmogorov's distance between the two distributions.

The last result deals with the multistable Lévy process with independent increments L₂ under the alternative hypothesis.

Theorem 3 (Alternative hypothesis for L₂). Assume that the observed process is L₂ (i.e.

X =L₂), that (^{n(N) log}_N ^N)_N is bounded and n(N) ≥ 256. Then there exists a constant K^∗ > 0 such that for allα ∈Θ₁ and x >0,

Pα(T_N ≤x)≤e^K∗

√x(K^∗)

√

n(N)e⁻

ξN b−a

b

R

a

|h(t)−h(t₀)|dt

. (21)

Thereby for a xed level β ∈ (0,1) in the testing problem, there exists a constant K_β > 0 such that for every α∈Θ1,

P^α(TN ∈Rc)≥1−K

√

n(N)

β e⁻

ξN b−a

b

R

a

|h(t)−h(t0)|dt

. (22)

Comparing with Theorem 2, we see that the power is asymptotically greater if X = L₂ because we can take advantage of less correlated data in that case.

5 Technical lemmas

We set in this section a list of lemmas used in the proofs of the main theorems. The rst one is a combinatory result, given without the proof.

Lemma 4. For every sequence (W_k,j)_(k,j)∈Z²,

bN bc−1

X

j=bN ac+1

j+^n(N)₂

X

k=j−^n(N)₂

W_k,j =

bN ac+^n(N)₂ −1

X

k=bN ac+1

k+^n(N)₂

X

j=bN ac+1

W_k,j+

bN bc−^n(N)₂ −1

X

k=bN ac+^n(N)₂

k+^n(N)₂

X

j=k−^n(N)₂ +1

W_k,j

+

bN bc+^n(N)₂ −2

X

k=bN bc−^n(N)

2

bN bc−1

X

j=k−^n(N)

2 +1

W_k,j.

The next lemma provides a uniform upper bound of the moment of the logarithm of standard α-stable random variables.

Lemma 5. Letη >0 and [α∗, α^∗]⊂(1,2). sup

α∈[α∗,α^∗]E^α[|log|W| −E^α[log[|W|]|^η]<+∞ (23) where W is a standard symmetric α-stable random variable.

In the next lemma, we provide a uniform upper bound of the moments of the inverse of the increments for the three processes L₁,L₂ and U.

Lemma 6. For all η∈(0,¹₂), sup

N

sup

k=1,...,N

sup

α∈Θ₁

N^−2h(tk)ηEα[|X_k,N|^−2η]<+∞ (24) where X =L₁, X =L₂ or X =U, and h(t) = _α(t)¹ .

We set now a control of the variances of ˆh^U_N(t) and ˆh_N(t).

Lemma 7. Assume thatX =L₁. For all η∈(0,¹₂), sup

N

sup

α∈Θ₁

sup

t∈[a,b]Eα

h

ξ_N²|¯h(t)−ˆh^U_N(t)|²i

<+∞, (25)

and

sup

N

sup

α∈Θ1

sup

t∈[a,b]Eα

h

N^{(1−α∗1 )η}|hˆ_N(t)−ˆh^U_N(t)|²i

<+∞. (26)

The last lemma states that the estimator of the functionh has exponential moments if the observed multistable process isL₂.

Lemma 8. Assume thatX =L₂ and (^{n(N) log}_N ^N)_N is bounded. Then

sup

N

sup

α∈Θ₁

sup

t∈[a,b]

bN tc+^n(N)

2 −1

sup

k=bN tc−^n(N)₂

Eα[N^h(t)|X_k,N|]<+∞, (27) and if n(N)≥256,

sup

N

sup

α∈Θ1

sup

t∈[a,b]

logEα[e^2|WN(t)|]

pn(N) <+∞ (28) where W_N(t) =ξ_N(ˆh_N(t)−h(t)).

6 Proofs

Proof of Lemma 5 Let α ∈ [α∗, α^∗], W be a standard α-stable random variable, and put µ(α) = Eα[log|W|] = Γ⁰(1)(1−_α¹).

E^α[|log|W| −E^α[log[|W|]|^η] = Z +∞

0

P^α(|log|W| −µ(α)|> x^1/η)dx

≤1 +

+∞

Z

1

Pα(|W| ≥e^x1/ηe^µ(α))dx+

+∞

Z

1

Pα(|W| ≤e^−x1/ηe^µ(α))dx.

Since the characteristic function of W is Eα[e^iyW] = e^−|y|α, Parseval's inversion formula provides an upper bound for the last term of the right handside of the previous inequality.

Moreover, the Markov inequality for the rst term leads to

E^α[|log|W| −µ(α)|^η]≤1 +e^−µ(α)E^α[|W|]

+∞

Z

1

e^−x1/ηdx+ 1 π

+∞

Z

1

Z

R

e^−|y|αsin(e^−x1/ηe^µ(α)y)

y dydx

≤1 +e^−µ(α)E^α[|W|]

+∞

Z

1

e^−x1/ηdx+e^µ(α) π

+∞

Z

1

Z

R

(e^−|y|+e^−|y|2)e^−x1/ηdydx.

We obtain (23) sinceEα[|W|] = _π²Γ(1−_α¹)(see Samorodnitsky and Taqqu 1994, property 1.2.17) and α∈[α∗, α^∗].

Proof of Lemma 6 We begin the proof showing that for X =L₁, X =L₂ or X =U, sup

N

sup

k=1,...,N

Z

R

φ^X_N(N^h(tk)θ)dθ <∞ (29) whereφ^X_N is the characteristic function ofXk,N. Recall that tk = _N^k and consider rst the case X = L₂. We know from Falconer and Liu (2012) that the characteristic function of X_k,N is given by φ^X_N(θ) =e⁻

Rtk+1

tk |θ|^α(x)dx

. Then we use the fact that α≥1 to obtain Z

R

φ^X_N(N^h(tk)θ)dθ = Z

R

e⁻

Rtk+1

tk |N^h(tk)θ|^α(x)dx

dθ

≤2 + 2

+∞

Z

1

e^−θ

R_tk+1

tk N^h(tk)α(x)dx

dθ

≤2 + 2

Rtk+1

tk N^h(tk)α(x)dx

= 2 + 2

NRtk+1

tk N^{h(tk)α(x)−1}dx. If α ∈ Θ₁, the mean value theorem yields |h(t_k)α(x)−1| = |^α(x)−α(t_α(t ^k)

k) | ≤ ^kα_{N α}^0k∞

∗ ≤ _{N α}^M

∗ so N^{h(tk)α(x)−1} ≥N^{−M/(N α∗)} and

Z

R

φ^X_N(N^h(tk)θ)dθ≤1 + 2 sup

N

N^{M/(N α∗)},

hence (29). We consider now the caseX =L1. Fort∈[a, b], recall thatC_α(t) = R∞

0 x^−α(t)sinx dx^−1/α(t) and write

A_k= C_α(tk+1)

2y^1/α(tk+1)1_[tk,tk+1](x) and

B_k = ( C_α(tk+1)

2y^1/α(tk+1) − C_α(tk)

2y^1/α(tk))1_[0,tk](x).

We know from Le Guével and Lévy-Véhel (2012) that the characteristic function of Xk,N = L₁(t_k+1)−L₁(t_k) is given by

E[e^iθXk,N] =e

−2

1

R

0 +∞

R

0

sin²(θ(Ak+Bk))dxdy

. It follows from the equality R∞

0

sin²(z)

z^1+α dz = ^2α−1_α C_α^−α that Z

R

φ^X_N(N^h(tk)θ)dθ ≤2 + 2

+∞

Z

1

e

−2

tk+1

R

tk +∞

R

0

sin²(N^h(tk)θAk)dxdy

dθ

= 2 + 2

+∞

Z

1

e^{−θα(tk+1)}dθ

≤2 + 2

+∞

Z

1

(e^−θ+e^−θ2)dθ,

which implies (29). Finally, the characteristic function ofUk,N isφ^U_N(θ) =e^−|θ|

α(tk)

N so we derive (29) since α∈ [α∗, α^∗].

Now we consider that X is eitherL₁, L₂ orU. Parseval's inversion formula yields Pα(|X_k,N| ≤y) = 1

π Z

R

sin(ξy)

ξ φ^X_N(ξ)dξ

= 1 π

Z

R

sin(N^h(tk)θy)

θ φ^X_N(N^h(tk)θ)dθ

≤ N^h(tk) π y

Z

R

φ^X_N(N^h(tk)θ)dθ

≤KN^h(tk)ysup

N

sup

k=1,...,N

Z

R

φ^X_N(N^h(tk)θ)dθ.

We obtain to conclude for η∈(0,¹₂)

E^α

|X_k,N|^−2η

=

∞

Z

0

P^α(|X_k,N| ≤N^−h(tk)v^−2η1 )N^2h(tk)ηdv

≤N^2h(tk)η(1 +

∞

Z

1

Pα(|X_k,N| ≤N^−h(tk)v^−2η1 )dv) and

sup

α∈Θ1

N^−2h(tk)ηE^α[|Xk,N|^−2η]≤1 +Ksup

N

sup

k=1,...,N

Z

R

φ^X_N(N^h(tk)θ)dθ

∞

Z

1

v^−2η1 dv which is (24).

Proof of Lemma 7 Let α∈Θ₁ and t∈[a, b]. From the denition (9) of ˆh^U_N(t),we have

¯h(t)−ˆh^U_N(t) = 1 n(N)

bN tc+^n(N)₂ −1

X

k=bN tc−^n(N)₂

log|Uk,N| −E^α[log|Uk,N|]

Γ⁰(1) + logN . We use the independence of the random variables(U_k,N)_k to obtain

var_α(ˆh^U_N(t)) = 1 ξ_N²

bN tc+^n(N)₂ −1

X

k=bN tc−^n(N)₂

v(α(t_k))≤ v(1) ξ_N² that is (25). To prove (26), write for α∈Θ₁ and t∈[a, b],

ˆh_N(t)−ˆh^U_N(t) = 1 n(N)

bN tc+^n(N)

2 −1

X

k=bN tc−^n(N)

2

log

U_k,N X_k,N whereby

Eα

h|ˆh_N(t)−ˆh^U_N(t)|²i¹₂

≤ 1 n(N)

bN tc+^n(N)₂ −1

X

k=bN tc−^n(N)₂

Eα

|log|Uk,N

X_k,N||^{2 1}₂

. (30)

Let η ∈ (0,¹₂) and x K_η > 0 such that for all x ∈ R, (logx)²1x≥1 ≤ K_η|x−1|^η. We deduce from the inequality

|log|x||² =|logx|²1x≥1+|log 1 x|²1¹

x≥1

≤K_η(|x−1|^η+|1

x −1|^η) that |log|_X^Uk,N

k,N||² ≤Kη|Uk,N −Xk,N|^η(|_X¹

k,N|^η +|_U¹

k,N|^η) which implies

E^α

|log|U_k,N X_k,N||²

≤KηE^α

|Uk,N −Xk,N|^2η1₂ E^α

(|Xk,N|^−η +|Uk,N|^−η)²¹₂

≤K_ηEα[|U_k,N−X_k,N|]^η Eα

|X_k,N|^−2η1₂ +Eα

|U_k,N|^−2η1₂

≤K_η^∗N^h(tk)ηEα[|U_k,N−X_k,N|]^η, (31) where we have used (24) for the last inequality. Let us controlEα[|U_k,N−X_k,N|]. To this end, thanks to the denition ofL₁ and (6), we get

X_k,N−U_k,N =

∞

X

i=1

γ_i

C_α(tk+1)Γ^−1/α(t_i ^k+1)−C_α(tk)Γ^−1/α(t_i ^k)

1_[0,tk+1](V_i).

The mean value theorem yields that there exists a random variables ci ∈ (α(tk), α(tk+1)) (or c_i ∈(α(t_k+1), α(t_k))) such that

C_α(tk+1)Γ^−1/α(t_i ^k+1)−C_α(tk)Γ^−1/α(t_i ^k) = (α(t_k+1)−α(t_k)) d

du(C_uΓ^−1/u_i )(c_i).

Note thatc_i is independent of γ_i. Then

E^α[|Xk,N −Uk,N|]≤ |α(tk+1)−α(tk)|E^α[|

∞

X

i=1

γi

d

du(CuΓ^−1/u_i )(ci)1_[0,tk+1](Vi)|.

Letp= 1 + ^α₂^∗ ∈(α^∗,2).We use the fact that u7→C_u is uniformly bounded which implies that there existsK >0such that|_du^d(C_uΓ^−1/u_i )(c_i)| ≤K(1 +|log Γ_i|)(Γ^−1/α_i ^∗+ Γ^−1/α_i ^∗)to obtain

Eα[|

∞

X

i=1

γ_i d

du(C_uΓ^−1/u_i )(c_i)1_[0,tk+1](V_i)|]≤KEα[(1 +|log Γ₁|)(Γ^−1/α₁ ^∗+ Γ^−1/α₁ ^∗)]

+KEα[|

∞

X

i=2

γ_i d

du(C_uΓ^−1/u_i )(c_i)1_[0,tk+1](V_i)|].

Moreover, sincec_i is independent of γ_i, Theorem 2 of Von Bahr and Essen (1965) entails Eα[|

∞

X

i=1

γ_i d

du(C_uΓ^−1/u_i )(c_i)1_[0,tk+1](V_i)|]≤KEα[(1 +|log Γ₁|)(Γ^−1/α₁ ^∗+ Γ^−1/α₁ ^∗)]

+K

∞

X

i=2

Eα[| d

du(C_uΓ^−1/u_i )(c_i)|^p]^1/p

whereby Eα[|

∞

X

i=1

γ_i d

du(C_uΓ^−1/u_i )(c_i)1_[0,tk+1](V_i)|]≤KEα[(1 +|log Γ₁|)(Γ^−1/α₁ ^∗ + Γ^−1/α₁ ^∗)]

+K

∞

X

i=2

Eα[|(1 +|log Γ_i|)(Γ^−1/α_i ^∗+ Γ^−1/α_i ^∗)|^p]^1/p

which is nite. Gathering all these inequalities, we obtain

Eα[|X_k,N −U_k,N|]≤K^∗(|α(t_k+1)−α(t_k)| ≤ K^∗M N

for some K^∗ >0. We derive from this last inequality and (31) that N^{(1−α∗1 )η}E^α h

|log|_X^Uk,N

k,N||²i is uniformly bounded (in k and N), and by (30), N^{(1−α∗1 )η}Eα

h|ˆh_N(t)−ˆh^U_N(t)|²i

is uniformly bounded in t∈[a, b], α∈Θ₁ and N, that is (26).

Proof of Lemma 8 Let us control Pα(|X_k,N| > y) for any y > 1. We write c_k,N = min

x∈[t_k,t_k+1]α(x). Using the truncation inequality (Loeve 1977, p. 209), one easily computes

Pα(|X_k,N|> y)≤y

1/y

Z

0

(1−e⁻

Rtk+1

tk |θ|^α(u)du

)dθ

≤y

1/y

Z

0 tk+1

Z

t_k

|θ|^α(u)dudθ

≤y

1/y

Z

0

|θ|^ck,N N dθ

≤ 1

2N y^ck,N. This leads to

Eα

N^h(t)|X_k,N|

≤1 + N^h(t)ck,N−1 2

+∞

Z

1

dy y^ck,N

≤1 + (

+∞

Z

1

dy

y^α∗)N^{ck,Nα(t)−1}. Thanks to the inequality |^c_α(t)^k,N −1| ≤ ^2kα0_{N α}^k∞n(N)

∗ whenever k ∈ {bN tc − ^n(N)₂ , ...,bN tc+

n(N)

2 −1},we obtain Eα

N^h(t)|X_k,N|

≤1 + (

+∞

Z

1

dy y^α∗) sup

N

e2M n(N) logN N

which is nite under the assumptions of the Lemma. This being true for everyN ≥1, t∈[a, b], α∈Θ₁, and k ∈ {bN tc − ^n(N)₂ , ...,bN tc+ ^n(N₂ ⁾−1}, (27) holds.

In order to prove (28), notice thatW_N(t) =p

n(N)µ(α(t))−√¹

n(N)

bN tc+^n(N)₂ −1

P

k=bN tc−^n(N)

2

log(|N^h(t)X_k,N|).

We use the following inequality for t∈[a, b]

e^2|WN(t)| ≤e^2WN(t)+e^−2WN(t)

=e²

√

n(N)µ(α(t))

bN tc+^n(N)₂ −1

Y

k=bN tc−^n(N)

2

1

|N^h(t)X_k,N|

√2 n(N)

+e⁻²

√

n(N)µ(α(t))

bN tc+^n(N)₂ −1

Y

k=bN tc−^n(N)

2

|N^h(t)X_k,N|

√2 n(N).

Taking the expectation, the independence of the variablesX_k,N and the Hölder inequality for pn(N)≥16 lead to

Eα[e^2|WN(t)|]≤(K^∗)

√

n(N)(

bN tc+^n(N)

2 −1

Y

k=bN tc−^n(N)

2

Eα[ 1

|N^h(t)Z_k,N|

√2 n(N)

] +

bN tc+^n(N)

2 −1

Y

k=bN tc−^n(N)

2

Eα[|N^h(t)Z_k,N|

√2 n(N)])

≤(K^∗)

√

n(N)(

bN tc+^n(N)

2 −1

Y

k=bN tc−^n(N)

2

Eα[|N^h(t)Z_k,N|⁻¹⁸]

√16 n(N) +

bN tc+^n(N)

2 −1

Y

k=bN tc−^n(N)

2

Eα[|N^h(t)Z_k,N|]

√2 n(N))

= (K^∗)

√

n(N)

bN tc+^n(N)

2 −1

Y

k=bN tc−^n(N)₂

N

2(h(tk)−h(t))

√

n(N) E^α[|N^h(tk)Z_k,N|⁻¹⁸]

√16 n(N)

+ (K^∗)

√

n(N)

bN tc+^n(N)₂ −1

Y

k=bN tc−^n(N)₂

E^α[|N^h(t)Zk,N|]

√2 n(N))

whereK^∗ is some positive constant. Thanks to (24), (27), and the inequality |h(tk)−h(t)| ≤

M n(N)

N whenever k ∈ {bN tc − ^n(N)₂ , ...,bN tc+ ^n(N₂ ⁾−1}, we obtain Eα[e^2|WN(t)|]≤(K^∗)

√

n(N)e^2M(n(N))3

/2 logN

N + (K^∗)

√

n(N)

≤(K^∗)

√

n(N)(sup

N

e2M n(N) logN

N )

√

n(N)+ (K^∗)

√

n(N)

which implies (28).

Proof of theorem 1 If α ∈ Θ₀, we can rewrite for all t ∈ [a, b], h(t) = h= _α¹. Let W_N(t) beξ_N(ˆh_N(t)−h) and recall that

T_N = 1 ˆ

σ²(t₁)(b−a) Z b

a

|W_N(t)−W_N(t₀)|²dt

= 1

ˆ

σ²(t₁)(b−a) Z b

a

|W_N(t)|²dt−2 W_N(t₀) ˆ

σ²(t₁)(b−a) Z b

a

W_N(t)dt+ 1 ˆ

σ²(t₁)|W_N(t₀)|².