Jean-Marc Bardet
email: bardet@univ-paris1.fr

Donatas Surgailis
email: sdonatas@ktl.mii.lt

Measuring the roughness of random paths by increment ratios

Keywords: AMS 2000 subject classifications: Primary 62H12, 60G17, Secondary 62F12, 60G18, 60G15, 60G51 Estimation of the local regularity function of stochastic process, Limit theorems, Hölder exponent, Tangent process, Zero crossings, Diffusion processes, Fractional Brownian motion, Multifractional Brownian motion, Lévy processes

A statistic based on increment ratios (IR) and related to zero crossings of increment sequence is defined and studied for measuring the roughness of random paths. The main advantages of this statistic are robustness to smooth additive and multiplicative trends and applicability to infinite variance processes. The existence of the IR statistic limit (called the IR-roughness below) is closely related to the existence of a tangent process. Three particular cases where the IR-roughness exists and is explicitly computed are considered. Firstly, for a diffusion process with smooth diffusion and drift coefficients, the IR-roughness coincides with the IR-roughness of a Brownian motion and its convergence rate is obtained. Secondly, the case of rough Gaussian processes is studied in detail under general assumptions which do not require stationarity conditions. Thirdly, the IR-roughness of a Lévy process with α-stable tangent process is established and can be used to estimate the fractional parameter α ∈ (0, 2) following a central limit theorem.

Introduction and the main results

It is well-known that random functions are typically "rough" (non-differentiable), which raise the question of determining and measuring roughness. Probably, the most studied roughness measures are the Hausdorff dimension and the p-variation index. There exists a considerable literature on statistical estimation of these and related quantities from a discrete grid. Hence, different estimators of the Hausdorff dimension have been studied, as the box-counting estimator (see [START_REF] Hall | On the performance of box-counting estimators of fractal dimension[END_REF] for stationary Gaussian processes or Lévy-Véhel and Peltier, 1994, for Gaussian processes with stationary increments). To our knowledge, the H-variation estimator, where H is a measurable function, was first proposed by [START_REF] Guyon | Convergence en loi des H-variations d'un processus gaussien stationnaire[END_REF] for stationary Gaussian processes where central and non-central limit theorems are established following the Hermite rank of H and the asymptotic local properties of the variogram and its second derivative. Further studies provided a continuation of this seminal paper in different ways. [START_REF] Istas | Quadratic variations and estimation of the local Hölder index of a Gaussian process[END_REF] studied generalized quadratic variations of Gaussian processes with stationary increments. [START_REF] Coeurjolly | Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths[END_REF][START_REF] Coeurjolly | Identification of multifractional Brownian motion[END_REF] studied ℓ p -variations of fractional Brownian motion and ℓ 2 -variations of multifractional Brownian motion. [START_REF] Coeurjolly | Hurst exponent estimation of locally self-similar Gaussian processes using sample quantiles[END_REF] discussed L-variations based on linear combinations of empirical quantiles for Gaussian locally self-similar processes. An estimator counting the number level crossings was investigated by [START_REF] Feuerverger | Estimation of fractal index and fractal dimension of a Gaussian process by counting the number of level crossings[END_REF] for stationary Gaussian processes.

In the present paper we introduce a new characteristic of roughness, defined as a sum of ratios of consecutive increments. For a real-valued function f = (f (t), t ∈ [0, 1]), define recursively

∆ 1,n j f := f j + 1 n -f j n , ∆ p,n j f := ∆ 1,n j ∆ p-1,n j f = p i=0 (-1) p-i p i f j + i n , (1.1)
so that ∆ p,n j f denotes the p-order increment of f at j n , p = 1, 2,

• • • , j = 0, 1, • • • , n -p. Let R p,n (f ) := 1 n -p n-p-1 k=0 ∆ p,n k f + ∆ p,n k+1 f |∆ p,n k f | + |∆ p,n k+1 f | , (1.2)
with the convention 0 0 := 1. In particular,

R 1,n (f ) = 1 n -1 n-2 k=0 f k+1 n -f k n + f k+2 n -f k+1 n f k+1 n -f k n + f k+2 n -f k+1 n . (1.3)
Note the ratio on the right-hand side of (1.2) is either 1 or less than 1 depending on whether the consecutive increments ∆ p,n k f and ∆ p,n k+1 f have same signs or different signs; moreover, in the latter case, this ratio generally is small whenever the increments are similar in magnitude ("cancel each other"). Clearly, 0 ≤ R p,n (f ) ≤ 1 for any f, n, p. Thus if lim R p,n (f ) exists when n → ∞, the quantity R p,n (f ) can be used to estimate this limit which represents the "mean roughness of f " also called the p-th order IR-roughness of f below. We show below that these definitions can be extended to sample paths of very general random processes, e.g. stationary processes, processes with stationary and nonstationary increments, and even L qprocesses with q < 1.

Let us describe the main results of this paper. Section 2 derives some general results on asymptotic behavior of this estimator. Proposition 2.1 says that, for a sufficiently smooth function f , the limit lim n→∞ R p,n (f ) = 1. In the most of the paper, f = X is a random process. Following [START_REF] Dobrushin | Automodel generalized random fields and their renorm group[END_REF], we say that X = (X t , t ∈ R) has a small scale limit Y (t0) at point t 0 ∈ R if there exist a normalization A (t0) (δ) → ∞ when δ → 0 and a random process Y (t0) = (Y (t0) τ , τ ≥ 0) such that

A (t0) (δ) (X t0+τ δ -X t0 ) f.d.d. -→ δ→0 Y (t0) τ , (1.4)
where f.d.d.

-→ stands for weak convergence of finite dimensional distributions. A related definition is given in [START_REF] Falconer | Tangent fields and the local structure of random fields[END_REF][START_REF] Falconer | The local structure of random processes[END_REF] who called the limit process Y (t0) a tangent process (at t 0 ). See also [START_REF] Benassi | Gaussian processes and pseudodifferential elliptic operators[END_REF]. In many cases, the normalization A (t0) (δ) = δ H(t0) , where 0 < H(t 0 ) < 1 and the limit tangent process Y (t0) is self-similar with index H(t 0 ) [START_REF] Falconer | The local structure of random processes[END_REF][START_REF] Dobrushin | Automodel generalized random fields and their renorm group[END_REF]. Proposition 2.2 states that if X satisfies a similar condition to (1.4), then the statistic R p,n (X) converges to the integral R p,n (X)

P -→ n→∞ 1 0 E |∆ p 0 Y (t) + ∆ p 1 Y (t) | |∆ p 0 Y (t) | + |∆ p 1 Y (t) | dt, (1.5)
in probability, where ∆ p j Y (t) = ∆ p,1 j Y (t) = p i=0 (-1) p-i p i Y (t)

j+i , j = 0, 1 is the corresponding increment of the tangent process Y (t) at t ∈ [0, 1). In the particular case when X has stationary increments, relation (1.5) becomes R p,n (X)

P -→ n→∞ E |∆ p 0 Y + ∆ p 1 Y | |∆ p 0 Y | + |∆ p 1 Y | . (1.6)
Section 3 discusses the convergence in (1.5) for diffusion processes X admitting a stochastic differential dX = a t dB(t) + b t dt, where B is a standard Brownian motion and (a t ), (b t ) are random (adapted) functions.

It is clear that under general regularity conditions on the diffusion and drift coefficients (a t ), (b t ), the process X admits the same local Hölder exponent as B at each point t 0 ∈ (0, 1) and therefore the IR-roughness of X in (1.5) should not depend on these coefficients and should coincide with the corresponding limit for X = B. This is indeed the case since the tangent process of X at t is easily seen to be Y (t) = a t B and the multiplicative factor a t cancels in the numerator and the denominator of the fraction inside the expectation in (1.5). See Proposition 3.1 for details, where the convergence rate O(n 1/3 ) (a.s.) in (1.5) with explicit limit values Λ p (1/2) is established for diffusions X and p = 1, 2.

Considerable attention is given to the asymptotic behavior of the statistic R p,n (X) for "fractal" Gaussian processes (see Section 4). In such a frame, fractional Brownian motion (fBm in the sequel) is a typical example. Indeed, if X is a fBm with parameter H ∈ (0, 1), then X is also its self tangent process for any t ∈ [0, 1] and (see Section 4): R p,n (X)

a.s. -→ n→∞ Λ p (H) (p = 1, 2), (1.7)
where Λ p (H) := λ(ρ p (H)), (1.8) λ(r) := 1 π arccos(-r) + 1 π

1 + r 1 -r log 2 1 + r , (1.9)
ρ p (H) := corr ∆ p 0 B H , ∆ p 1 B H , (1.10)
and where ∆ The difference in the range of the parameter H for p = 1 and p = 2 in the central limit theorem in (1.11) are due to the fact that the second order increment process ∆ 2 j B H , j ∈ Z is a short memory stationary Gaussian process for any H ∈ (0, 1), in contrast to the first order increment process ∆ 1 j B H , j ∈ Z which has long memory for H > 3/4. Generalizations of (1.7) and (1.11) to Gaussian processes having nonstationary increments are proposed in Section 4. Roughly speaking, R p,n (X), p = 1, 2 converge a.s. and satisfy a central limit theorem, provided for any t ∈ [0, 1] the process X admits a fBm with parameter H(t) as a tangent process (more precise assumptions (A.1), (A.1) ′ and (A.2) p are provided in Section 4). In such frames, the limits in (1.7) are 1 0 Λ p (H(t))dt instead of Λ p (H) and the asymptotic variances in (1.11) also change. The case of Gaussian processes with stationary increments is discussed in detail and the results are used to define a √ n-consistent estimator of H, under semiparametric assumptions on the asymptotic behavior of the variogram or the spectral density. [START_REF] Bardet | Nonparametric estimation of the local Hurst function of multifractional processes[END_REF] study a punctual estimator of H(t 0 ) obtained from a localization around t 0 ∈ (0, 1) of the statistic R 2,n (X).

1 j B H = B H (j + 1) -B H (j), ∆ 2 j B H = B H (j + 2) -2B H (j + 1) -B H (j) (j ∈ Z) are respective increments of fBm. Moreover, √ n R p,n (X) -Λ p (H) D -→ n→∞ N (0, Σ p (H)) if p = 1, 0 < H < 3/4, p = 2, 0 < H < 1, (1.11) where
The main advantages of estimators of the type (1.2) involving a scaling invariant function of increments seem to be the following. Firstly, the estimator R p,n (X) essentially depends on local regularity of the process X and not on possible "multiplicative and additive factors" such as diffusion and drift coefficients in Section 3 or smoothly multiplicative and additive trended Gaussian processes, see Proposition 4.1 of Section 4. This property is important when dealing with financial data involving heteroscedasticity and volatility clustering. Such a robustness property (also satisfied by the estimators based on generalized quadratic variations of wavelet coefficients) represents a clear advantage versus classical parametric Whittle or semi-parametric logperiodogram estimators. Secondly, the estimators in (1.2) are bounded functionals and have finite moments of any order. Section 5 discusses jump Lévy processes, with the Lévy measure regularly varying of fractional index α ∈ (0, 2) at the origin. Using a modification of (1.2), we define a √ n-consistent estimator of α, together with a central limit theorem, in a very general semiparametric frame. This result is new and interesting because there exist very few papers providing consistent estimators of α (to our knowledge, the only comparable results have been established in [START_REF] Belomestny | Spectral estimation of the fractional order of a Lévy process[END_REF] and (Ait Sahalia and Jacod, 2009) in a financial and somewhat different context). Finally, in the Gaussian case, using the approximation formula provided in Remark 4.3, an estimator of H based on R 2,n (X) can be extremely simply computed:

H (2) n ≃ 1 0.1468 1 n -2 n-3 k=0 X k+2 n -2X k+1 n + X k n + X k+3 n -2X k+2 n + X k+1 n X k+2 n -2X k+1 n + X k n + X k+3 n -2X k+2 n + X k+1 n . -0.5174 . In the R language, if X is the vector X 1 n , X 2 n , • • • , X 1 , H (2) n ≃ (mean(abs(diff(diff(X[-1])) + diff(diff(X[-length(X)]))) /(abs(diff(diff(X[-1]))) + abs(diff(diff(X[-length(X)]))))) -0.5174)/0.1468.
Therefore its computation is very fast and does not require any tuning parameters such as the scales for estimators based on quadratic variations or wavelet coefficients. The convergence rate of our estimator is √ n as for the parametric Whittle or the generalized quadratic variation estimators and hence it is more accurate than most of other well-known semi-parametric estimators (log-periodogram, local Whittle or wavelet based estimators).

Estimators of the form (1.2) can also be applied to discrete time (sequences) instead of continuous time processes (functions). For instance [START_REF] Surgailis | The increment ratio statistic[END_REF] extended the statistic R 2,n (X) to discrete time processes and used it to test for I(d) behavior (-1/2 < d < 5/4) of observed time series. [START_REF] Vaičiulis | An estimator of the tail index based on increment ratio statistics[END_REF] considered estimation of the tail index of i.i.d. observations using an increment ratio statistic.

Remark 1.1 The referee noted that the IR-roughness might be connected to the level crossing index (see [START_REF] Feuerverger | Estimation of fractal index and fractal dimension of a Gaussian process by counting the number of level crossings[END_REF]. To our surprise, such a connection indeed exists as explained below. Let Y n (t), t ∈ [0, 1-1 n ] be the linear interpolation of the "differenced" sequence ∆

1,n j X = X( j+1 n )-X( j n ), j = 0, 1, • • • , n-1: Y n (t) = n ( j + 1 n -t)∆ 1,n j X + (t - j n )∆ 1,n j+1 X , t ∈ [ j n , j + 1 n ), j = 0, 1, • • • , n -2.
Then, using Figure 3 as a proof,

R 1,n (X) = n n -1 n-2 j=0 meas t ∈ [ j n , j + 1 n ) : Y n (t) > 0 -meas t ∈ [ j n , j + 1 n ) : Y n (t) < 0 (1.13) = n n -1 n-2 j=0 j+1 n j n (1(Y n (t) > 0) -1(Y n (t) < 0))dt . J J J J J J J J J J J J J J J J J J t Y n ( j n ) j n j+1 n Y n ( j+1 n ) U2 U1 1 Figure 3. The proof of (1.13): follows by |Yn( j n )+Yn( j+1 n )| |Yn( j n )|+|Yn( j+1 n )| = n|U 1 -U 2 |. Let ψ(x 1 , x 2 ) := |x 1 + x 2 |/(|x 1 | + |x 2 |), ψ 0 (x 1 , x 2 ) := 1(x 1 x 2 ≥ 0). Clearly, the two quantities 1 - ψ Y n ( j n ), Y n ( j+1 n ) and 1 -ψ 0 Y n ( j n ), Y n ( j+1 n
) both are strictly positive if and only if Y n crosses the zero level in the interval [ j n , j+1 n ) but the former quantity measures not only the fact but also the "depth" of the crossing so that 1ψ Y n ( j n ), Y n ( j+1 n ) attains its maximal value 1 in the case of a "perfect" crossing in the middle of the interval [ j n , j+1 n ) (see Figure 3). It seems that similar asymptotic results can be obtained for R p,n 0 (X) := 1 n-p n-p-1 k=0 ψ 0 ∆ p,n k X, ∆ p,n k+1 X measuring the number of zero crossings of the increment sequence ∆ p,n k X, k = 0, 1, • • • , np and other similar statistics obtained by replacing the functions ψ or ψ 0 by other scaling invariant functions. Let us note that R 1,n 0 (X) is related to the zero-crossings' counting statistic studied in Ho and Sun (1987) for stationary Gaussian time series. Also note that the Hermite rank of ψ 0 is 2 and that the corresponding limit function λ 0 (r) = 1 π arccos(-r) is strictly increasing on the interval (-1, 1) similarly as the function λ(r) in (1.9). On the other hand, while the statistic R p,n 0 (X) is certainly of interest, the statistic R p,n (X) seems preferable to it for the reasons explained above. In particularly, in the case of symmetric Lévy processes X with independent increments studied in Section 5, the latter statistic leads to an estimator of the fractional index while the former statistic can be easily shown to converge to 1/2. The paper is organized as follows. Section 2 discusses some general (consistency) properties of the estimators R p,n (X). Section 3 deals with the case when X is a diffusion. The case of Gaussian processes X is considered in Section 4 while the case of Lévy processes is studied in Section 5. Section 6 contains proofs and other derivations.

Below, we write C for generic constants which may change from line to line.

Some asymptotic results

The definition of R p,n f in (1.2) can be extended to more general increments (the so-called generalized variations). Consider a filter a := (a 0 , • • • , a q ) ∈ R q+1 such that there exists p ∈ N, p ≤ q satisfying q ℓ=0 ℓ i a ℓ = 0 for i = 0, • • • , p -1 and

q ℓ=0 ℓ p a ℓ = 0. (2.1)
The class of such filters will be denoted A(p, q). For n

∈ N * := {1, 2, • • •} and a function f = (f (t), t ∈ [0, 1]), define the generalized variations of f by ∆ a,n j f := q ℓ=0 a ℓ f ( j + ℓ n ), j = 0, 1, • • • , n -q. (2.2)
A particular case of (2.2) corresponding to q = p ≥ 1, a ℓ = (-1) p-ℓ p ℓ is the p-order increment ∆ p,n j f in (1.1). For a filter a ∈ A(p, q), let

R a,n (f ) := 1 n -q n-q-1 k=0 ∆ a,n k f + ∆ a,n k+1 f |∆ a,n k f | + |∆ a,n k+1 f | . (2.
3)

It is easy to prove that R 1,n (f ) -→ n→∞ 1 if f is continuously differentiable on [0, 1] and the derivative f ′ does not vanish on [0, 1] except maybe for a finite number of points. Moreover, it is obvious that R 1,n (f ) = 1 if f is monotone on [0, 1]: the IR-roughness of a monotone function is the same as of a smooth function, which is not surprising since a similar fact holds for other measures of roughness such as the p-variation index or the Hausdorff dimension. We conjecture that R p,n (f ) → 1 and R a,n (f ) → 1 for any q ≥ p ≥ 1, a ∈ A(p, q) and f : [0, 1] → R which is (p -1) times differentiable and the derivative f (p-1) has bounded variation on [0, 1] with the support supp(f (p-1) ) = [0, 1]. However, we can prove a weaker result.

Proposition 2.1 Let f be (p -1)-times continuously differentiable (p ≥ 1) with f (p-1) being absolutely continuous on [0, 1] having the Radom-Nikodym derivative g = f (p-1) ′ . Assume that g = 0 a.e. in [0, 1]. Then R p,n (f ) -→ n→∞ 1 and R a,n (f ) -→ n→∞
1 for any a ∈ A(p, q), q ≥ p.

Proof. We restrict the proof to the case p = 2 since the general case is analogous. Using summation by parts, we can rewrite ∆ a,n j f as

∆ a,n j f = q i=0 b i ∆ 2,n i+j f, (2.4
)

where b i := i k=0 k ℓ=0 a ℓ , i = 0, 1, • • • , q, b q-1 = b q = 0 and b := q i=0 b i = 1 2 q i=1 i 2 a i = 0 in view of the assumption a ∈ A(2, q).
Assume n is large enough and for a given t ∈ (0, 1), let k n (t) ∈ {0, • • • , n -2} be chosen so that t ∈ [k n (t)/n, (k n (t) + 1)/n), therefore k n (t) = [nt] -1. We claim that for a.e. t ∈ (0, 1)

lim n→∞ n 2 ∆ a,n kn(t) f = b g(t), lim n→∞ n 2 ∆ a,n kn(t)+1 f = b g(t). (2.5)
Using the fact that the function (x 1 , x 2 ) → |x1+x2| |x1|+|x2| is continuous on R 2 \{(0, 0)}, we obtain

h a,n (t) := n 2 ∆ a,n kn(t) f + n 2 ∆ a,n kn(t)+1 f n 2 ∆ a,n kn(t) f + n 2 ∆ a,n kn(t)+1 f -→ n→∞ | b g(t) + b g(t)| | b g(t)| + | b g(t)| = 1 (2.6)
for a.e. t ∈ (0, 1), where we used the fact that b g(t) = 0 a.e. Since for n ≥ q, R a,n (f ) can be written as

R a,n (f ) = n n-q 1 0 h a,n (t)dt, relation R a,n (f ) -→ n→∞ 1
follows by the dominated convergence theorem and the fact that 0 ≤ h a,n (t) ≤ 1. Relations (2.5) can be proved using the Lebesgue-Vitali theorem (see [START_REF] Shilov | Integral, Measure and Derivative[END_REF], Ch. 4, §10, Theorem 1), as follows. Consider the signed measure µ on Borel subsets of [0, 1/2] 2 given by

µ(A) = A g(x 1 + x 2 )dx 1 dx 2 . Note ∆ 2,n k f = µ((k/2n, (k + 2)/2n] × (k/2n, (k + 2)/2n]), (k = 0, • • • , n -2). Since rectangles [x 1 , x 1 + h] × [x 2 , x 2 + h], (0 ≤ x i < x i + h ≤ 1/2, i = 1, 2
) form a Vitali system on [0, 1/2] 2 , the above mentioned Lebesgue-Vitali theorem implies that ], n = 1, 2, • • • is regularly contracting to (t 1 , t 2 ) ∈ (0, 1) 2 in the sense of (Shilov and Gurevich, 1967, Ch. 4, §10). Hence, using the lemma on p. 214 of the above monograph, it follows that n 2 µ kn(t1)+i 2n , kn(t1)+i+2

φ n (t 1 , t 2 ) := n 2 µ kn(t1) 2n , kn(t1)+2 2n × kn(t2) 2n , kn(t2)+2 2n -→ n→∞ g t1+t2 2 (2.7)
2n × kn(t2)+i 2n , kn(t2)+i+2 2n -→ n→∞ g t1+t2 2 a.e. in [0, 1] 2 , implying n 2 ∆ 2,n kn(t)+i f -→ n→∞ g(t)
a.e. on [0, 1], for any i = 0, 1, • • • .

Together with (2.4), this proves (2.5) and the proposition.

Let us turn now to the case when f (t) = X t , t ∈ [0, 1] is a random process. Now and in all the sequel, R p,n (X), R a,n (X) are denoted R p,n , R a,n , respectively. Below we formulate a general condition for the convergence of R p,n and R a,n to a deterministic limit.

Assumption (A): For a.e. pairs (t 1 , t 2 ) ∈ (0, 1) 2 , t 1 = t 2 , for i = 1, 2 there exist: (i) normalizations A (ti) (δ) → ∞ (δ → 0), (ii) (mutually) independent random processes Y (ti) = (Y (ti) (τ ), τ ∈ [0, 1]), such that for δ → 0, s 1 → t 1 , s 2 → t 2 A (t1) (δ)(X s1+δτ -X s1 ), A (t2) (δ)(X s2+δτ -X s2 ) f.d.d. -→ (Y (t1) (τ ), Y (t2) (τ )).
(2.8)

Remark 2.1 Relation (2.8) implies the existence of a joint small scale limit (Y (t1) , Y (t2) ) at a.e. pair (t 1 , t 2 ) ∈ (0, 1), with independent components Y (t1) , Y (t2) . Note Assumption (A) and Proposition 2.2 below are very general, in the sense that they do not assume any particular structure or distribution of X.

Proposition 2.2 Let a = (a 0 , • • • , a q ) ∈ A(p, q), 1 ≤ p ≤ q be a filter and let X satisfy Assumption (A). Assume in addition that

P(|∆ a j Y (t) | > 0) = 1, j = 0, 1 for a.e. t ∈ (0, 1), where ∆ a j Z ≡ ∆ a,1 j Z = q ℓ=0 a ℓ Z(j + ℓ). Then E R a,n - 1 0 E |∆ a 0 Y (t) + ∆ a 1 Y (t) | |∆ a 0 Y (t) | + |∆ a 1 Y (t) | dt 2 -→ n→∞ 0.
(2.9)

Proof. The statement follows from

ER a,n -→ n→∞ 1 0 E |∆ a 0 Y (t) + ∆ a 1 Y (t) | |∆ a 0 Y (t) | + |∆ a 1 Y (t) | dt and E (R a,n -ER a,n ) 2 -→ n→∞ 0. (2.10) Write ER a,n = n n-q 1 0 Eh a,n X (t)dt, where (c.f. (2.6)) h a,n X (t) := A (t) (1/n) ∆ a,n kn(t) X + ∆ a,n kn(t)+1 X A (t) (1/n) ∆ a,n kn(t) X + ∆ a,n kn(t)+1 X D -→ n→∞ |∆ a 0 Y (t) + ∆ a 1 Y (t) | |∆ a 0 Y (t) | + |∆ a 1 Y (t) | =: h a Y (t),
for a.e. t ∈ (0, 1), according to Assumption (A) and the continuous mapping theorem. Whence and from 0 ≤ h a,n X ≤ 1 using the Lebesgue dominated convergence theorem the first relation in (2.10) follows. Moreover,

E(R a,n -ER a,n ) 2 = n n -q 2 1 0 1 0 E h a,n X (t)h a,n X (t ′ ) dtdt ′ -ER a,n 2 ,
and with the same arguments as previously and the independence of

Y (t) and Y (t ′ ) when t = t ′ , E h a,n X (t)h a,n X (t ′ ) -→ n→∞ E |∆ a 0 Y (t) + ∆ a 1 Y (t) | |∆ a 0 Y (t) | + |∆ a 1 Y (t) | • |∆ a 0 Y (t ′ ) + ∆ a 1 Y (t ′ ) | |∆ a 0 Y (t ′ ) | + |∆ a 1 Y (t ′ ) | = Eh a Y (t) • Eh a Y (t ′ )
and therefore The following easy but interesting corollary can also be added to this result. It proves that smooth multiplicative or additive trends do no change the L 2 -asymptotic behavior of R a,n . Let C p [0, 1] denote the class of all p-times continuously differentiable functions on [0, 1].

n n-q 2 1 0 1 0 E h a,n X (t)h a,n X (t ′ ) dtdt ′ - 1 0 Eh a,n X (t)dt
Corollary 2.1 Let a ∈ A(p, q) and X satisfy the conditions of Proposition 2.2 with

A (t) (δ) = O(δ -1 ) (δ → 0) for each t ∈ [0, 1]. Assume that α ∈ C 1 [0, 1], β ∈ C p [0, 1], inf t∈[0,1] α(t) > 0 and sup t∈[0,1] |X(t)| < ∞ a.s. Define Z such that Z t = α(t) X t + β(t), t ∈ [0, 1]. Then (2.9) holds with R p,n = R p,n (X) replaced by R p,n (Z).
Proof. We consider here p ≥ 2 but the case p = 1 can be easily obtained. With α

( k+j n ) = α( k n ) + j n α ′ ( k n ) 1 + o(1) and ∆ a,n k β = O 1 n 2 , for a.e. t ∈ (0, 1) and k = k n (t) as defined in the proof of Proposition 2.1, we deduce that ∆ a,n k Z = α k n ∆ a,n k X + 1 n α ′ k n ∆ a ′ ,n k X + o 1 n sup t∈(0,1) |X(t)| + O 1 n 2 ,
with a ′ = (ja j ) 0≤j≤q ∈ A(p -1, q). Therefore,

A (t) ( 1 n )∆ a,n kn(t) Z = α k n (t) n A (t) 1 n ∆ a,n kn(t) X + 1 n O p A (t) 1 n ∆ a ′ ,n kn(t) X + o p 1 n A (t) ( 1 n ) D -→ n→∞ α(t)∆ a 0 Y (t) .
In a similar way, for a.e. pairs (t, t ′ ) ∈ (0, 1) 2 , t = t ′ , we can verify the joint convergence in distribution of r.v.'s

A (t) ( 1 n )∆ a,n kn(t) Z, A (t) ( 1 n )∆ a,n kn(t)+1 Z, A (t ′ ) ( 1 n )∆ a,n kn(t ′ ) Z, A (t ′ ) ( 1 n )∆ a,n kn(t ′ )+1 Z to the limiting r.v.'s α(t)∆ a 0 Y (t) , α(t)∆ a 1 Y (t) , α(t ′ )∆ a 0 Y (t ′ ) , α(t ′ )∆ a 1 Y (t ′ )
. Now, the statement of the corollary follows by the argument at the end of the proof of Proposition 2.2.

Remark 2.2 By definition, the statistics R p,n and R a,n for a ∈ A(p, q), 1 ≤ p ≤ q are invariant with respect to additive polynomial trends of order less than p; in particular, R 3,n is insensitive to a quadratic trend while R 2,n does not have this property. On the other hand, Corollary 2.1 (see also Proposition 4.1) says that under weak additional conditions on X, any sufficiently smooth additive or multiplicative trends do not affect the limit of R p,n as soon as p ≥ 1. In the important special case when the limit process Y (t) = B H in Assumption (A) and (2.9) is a fractional Brownian motion with parameter H ∈ (0, 1) independent of t, the statistic R p,n converges in mean square to the expectation E

|∆ p 0 BH +∆ p 1 BH | |∆ p 0 BH |+|∆ p 1 BH | = λ(ρ p (H)), c.f. (1.8)-(1.10
). Numerical computations show that the correlation coefficient ρ p (H) is a monotone function of H for any p ≥ 1 and tends to constant value -1 on the interval (0, 1) as p increases. Therefore, for larger values of p, the range of λ(ρ p (H)) is rather small and R p,n seems less capable to estimate H. A final reason for our concentrating on the "lower-order" statistics R p,n , p = 1, 2 in the rest of the paper is the fact that R 2,n satisfies the central limit theorem in (1.11) on the whole interval H ∈ (0, 1).

Diffusions

Let

X t = X 0 + t 0 a s dB(s) + t 0 b s ds, t ∈ [0, 1] (3.1)
be a diffusion (or Itô's) process on R. In (3.1), we assume the existence of a right-continuous filtration F = (F t , t ∈ [0, 1]), a standard Brownian motion B adapted to F ; moreover, a s , b s , s ∈ [0, 1] are adapted random functions satisfying

1 0 |b s |ds < ∞, 1 0 a 2 s ds < ∞ a.s., and X 0 is a F 0 -measurable r.v. Write E t [•] = E[•|F t ] for the conditional expectation. Let Λ 1 (1/2) = λ(ρ 1 (1/2)) ≃ 0.7206 and Λ 2 (1/2) = λ(ρ 2 (1/2)) ≃ 0.5881.
The proof of the following Lemma 3.1 is given in Annexe.

Lemma 3.1 Let ψ(x 1 , x 2 ) := |x 1 + x 2 |/(|x 1 | + |x 2 |) (x 1 , x 2 ∈ R)
, and let Z i , i = 1, 2 be independent N (0, 1) r.v.'s. Then for any random variables ξ 1 , ξ 2 ,

|Eψ(Z 1 + ξ 1 , Z 2 + ξ 2 ) -Eψ(Z 1 , Z 2 )| ≤ 20 max i=1,2 Eξ 2 i 1/3 . (3.2)
Theorem 3.1 Assume the following conditions: there exist random variables K 1 , K 2 such that 0 < K i < ∞ a.s., and such that, for any sufficiently small h > 0 and any 0 ≤ t < t + h ≤ 1, the following inequalities hold, a.s.:

|a t | ≥ K 1 , E t b 2 t+h ≤ K 2 and E t (a t+h -a t ) 2 ≤ K 2 h. (3.3) Then R p,n -Λ p (1/2) = O(n -1/3 ) a.s. (p = 1, 2). (3.4)
Proof. We restrict the proof to the case p = 1 since the case p = 2 is analogous. For notational simplicity, assume that n is odd. Define

η n (k) := |∆ 1,n k X + ∆ 1,n k+1 X| |∆ 1,n k X| + |∆ 1,n k+1 X| , η ′ n (k) := E k/n [η n (k)], η ′′ n (k) := η n (k) -η ′ n (k) (3.5)
and correspondingly write

R 1,n = R ′ n + R ′′ n , R ′ n := (n -1) -1 n-2 k=0 η ′ n (k), R ′′ n1 := (n -1) -1 (n-2)/2 k=0 η ′′ n (2k), R ′′ n2 := (n -1) -1 (n-4)/2 k=0 η ′′ n (2k + 1). As (η ′′ n (2k), F (2k+2)/n , k = 0, . . . , (n -2)/2
) is a martingale difference sequence, so by Burkholder's inequality,

E(R ′′ n1 ) 8 ≤ Cn -8   (n-2)/2 k=0 E 1/4 (η ′ n (2k)) 8   4 ≤ Cn -4
and therefore

∞ n=1 P(|R ′′ n1 | > n -1/3 ) ≤ C ∞ n=1 n 8/3 n -4 < ∞, implying R ′′ n1 = O(n -1/
3 ) a.s. A similar fact holds for R ′′ n2 . Thus, it remains to prove

R ′ n -Λ 1 (1/2) = O(n -1/3 ) a.s. (3.6) Observe η ′ n (k) -Λ 1 (1/2) = E k/n |Z 1 (k) + ξ 1 (k) + Z 2 (k) + ξ 2 (k)| |Z 1 (k) + ξ 1 (k)| + |Z 2 (k) + ξ 2 (k)| -E |Z 1 (k) + Z 2 (k)| |Z 1 (k)| + |Z 2 (k)| ,
where

Z 1 (k) := n 1/2 ∆ 1,n k B, Z 2 (k) := n 1/2 ∆ 1,n k+1 B, ξ 1 (k) := n 1/2 (k+1)/n k/n a s a k/n -1 dB(s) + n 1/2 (k+1)/n k/n b s a k/n ds, ξ 2 (k) := n 1/2 (k+2)/n (k+1)/n a s a k/n -1 dB(s) + n 1/2 (k+2)/n (k+1)/n b s a k/n ds. According to Lemma 3.1 above, |η ′ n (k) -Λ 1 (1/2)| ≤ 36 max i=1,2 E k/n ξ 2 i (k)
1/3 and therefore

|R ′ n -Λ 1 (1/2)| ≤ 36 max{(E k/n ξ 2 i (k)) 1/3 : i = 1, 2, k = 0, 1, . . . , n -1}.
Whence, (3.6) follows from the following fact: there exists a r.v. K < ∞, independent of n and such that for any n ≥ 1, k = 0, . . . , n -

1, i = 1, 2 E k/n ξ 2 i (k) ≤ K n -1 , a.s. (3.7) Indeed, using (3.3), E k/n ξ 2 1 (k) = n (k+1)/n k/n E k/n a s a k/n -1 2 ds + nE k/n (k+1)/n k/n b s a k/n ds 2 ≤ nK -2 1 (k+1)/n k/n E k/n a s -a k/n 2 ds + K -2 1 (k+1)/n k/n E k/n b 2 s ds ≤ K 2 K -2 1 n -1 , a.s.,
and the bound (3.7) for i = 2 follows similarly. This proves (3.7) and Theorem 3.1, too.

Let us present some examples of Itô's processes X satisfying conditions (3.3).

Example 3.1 Let (X t , t ∈ [0, 1]) be a Markov process satisfying a stochastic equation

X t = x 0 + t 0 a(X s )dB(s) + t 0 b(X s )ds, (3.8)
where x 0 ∈ R is nonrandom, a(x), b(x), x ∈ R are real measurable functions and B is a standard Brownian motion. Let F t := σ{B(s), s ≤ t}, 0 ≤ t ≤ 1 be the natural filtration. Assume that

|a(x) -a(y)| ≤ K|x -y|, |b(x) -b(y)| ≤ K|x -y| (x, y ∈ R) (3.9)
for some constant K < ∞. Then equation (3.8) admits a unique adapted solution; see e.g. [START_REF] Gikhman | Introduction to the Theory of Random Processes[END_REF]. Let a t = a(X t ), b t = b(X t ). Assume in addition that |a(x)| ≥ K 1 , (x ∈ R) for some nonrandom constant K 1 > 0. Then the first inequality in (3.3) is trivially satisfied; moreover, the second and third relations in (3.3) are also satisfied, with

K 2 = C(1 + sup 0≤t≤1 X 2 t ) < ∞ and K 3 = C,
|g x (t, x)| ≥ K 1 , |g x (s, y) -g x (t, x)| ≤ K(|s -t| 1/2 + |y -x|),
for all (t, x), (s, y) ∈ [0, 1] × R) and some constants 0 < K 1 , K < ∞. Then X satisfies the conditions in (3.3).

Gaussian processes 4.1. Assumptions

Let X = (X t , t ∈ [0, 1]) be a Gaussian process, with zero mean. Without loss of generality, assume X 0 = 0. Define σ 2 p,n (k), the variance of ∆ p,n k X, and ρ p,n (k), the correlation coefficient between ∆ p,n k X and ∆ p,n k+1 X, i.e.

σ 2 p,n (k) := E (∆ p,n k X) 2 , ρ p,n (k) := E ∆ p,n k X ∆ p,n k+1 X σ p,n (k)σ p,n (k + 1) . ( 4
B H ≡ ∆ 1 j B H = B H (j + 1) -B H (j) and ∆ 2 j B H = B H (j + 2) -2B H (j + 1) + B H (j) can be explicitly calculated: E ∆ 0 B H ∆ j B H = 2 -1 |j + 1| 2H + |j -1| 2H -2|j| 2H , (4.2) E ∆ 2 0 B H ∆ 2 j B H = 2 -1 -|j + 2| 2H + 4|j + 1| 2H -6|j| 2H + 4|j -1| 2H -|j -2| 2H . (4.3) From Taylor expansion, E ∆ 0 B H ∆ j B H ∼ 2H(2H -1)j 2H-2 , E ∆ 2 0 B H ∆ 2 j B H ∼ 2H(2H -1)(2H -2)(2H -3)j 2H-4 ,
as j → ∞, and therefore the first increment, (∆ j B H ), has a summable covariance if and only if 0 < H < 3/4, while the second increment, (∆ 2 j B H ), has a summable covariance for any 0 < H < 1.

Introduce the following conditions:

(A.
H(t) -H(t + 1 n ) √ n log n = 0 and lim n→∞ sup t∈(0,1) c(t) -c(t + 1 n ) √ n = 0. (4.7) (A.2) p There exist d > 0, γ > 1/2 and 0 ≤ θ < γ/2 such that for any 1 ≤ k < j ≤ n with n ∈ N * E ∆ p,n k X ∆ p,n j X ≤ d σ p,n (k)σ p,n (j) • n θ • |j -k| -γ . (4.8)
A straightforward application of Assumption (A.1) (or (A.1) ′ ) implies that c(t)B H(t) is the tangent process of X for all t ∈ (0, 1) and more precisely:

E ∆ p,n [nt] X ∆ p,n j+[nt] X E ∆ p,n 0 B H(t) ∆ p,n j B H(t) -c(t) = 0, (4.9)
lim n→∞ √ n sup t∈(0,1) E ∆ p,n [nt] X ∆ p,n j+[nt] X E ∆ p,n 0 B H(t) ∆ p,n j B H(t) -c(t) = 0. (4.10)
Moreover, for any t ∈ (0, 1) and p = 1, 2

n H(t) ∆ p,n j+[nt] X j∈Z f.d.d.
-→ n→∞ c(t) ∆ p j B H(t) j∈Z .

Assumption (A.1) can be characterized as uniform local self-similarity of (X t ) (the uniformity refers to the supremum over t ∈ (0, 1) in (4.4)). Note that for X having stationary increments and variogram V (t) = EX 2 t , Assumption (A.1) reduces to V (t) ∼ ct 2H (c > 0, 0 < H < 1). For j = 0, 1, relation (4.9) implies that for any t ∈ (0, 1), the variance and the (1/n)-lag correlation coefficient of ∆ p,n

[nt] X satisfy the following relations:

σ 2 1,n ([nt]) ∼ n→∞ c(t)σ 2 1 (H(t)) = c(t) E (∆ 0 B H(t) ) 2 = c(t) 1 n 2H(t) , (4.11
)

ρ 1,n ([nt]) -→ n→∞ ρ 1 (H(t)) = corr(B H(t) (1), B H(t) (2) -B H(t) (1)) = 2 2H(t)-1 -1, (4.12
) Condition (4.8) specifies a nonasymptotic inequality satisfied by the correlation of increments ∆ p,n k X. The particular case of stationary processes allows to better understand this point. Indeed, if (X t ) has stationary increments, the covariance of the stationary process (∆ p,n k X, k ∈ Z) is completely determined by the variogram V (t), e.g.

σ 2 2,n ([nt]) ∼ n→∞ c(t)σ 2 2 (H(t)) = c(t) E (∆ 2 0 B H(t) ) 2 = c(t)(4 -4 H(t) ) 1 n 2H(t) , (4.13)
ρ 2,n ([nt]) -→ n→∞ ρ 2 (H(t)) = corr(B H(t) (2) -2B H(t) , B H(t) (3) -2B H(t) (2) + B H(t) (1)) = -3 2H(t) + 2 2H(t)+2 -7 8 -2 2H(t
E ∆ 1,n k X ∆ 1,n j X = 1 2 V k -j + 1 n + V k -j -1 n -2V k -j n . (4.15)
In the "most regular" case, when X = B H is a fBm and therefore V (t) = t 2H , it is easy to check that assumption (A.2) 2 holds with θ = 0 and γ = 4 -2H > 2 (0 < H < 1), while (A.2) (ii) Assume, in addition, that |V (4) (t)| ≤ Ct -κ (0 < t < 1), for some C > 0 and 8 -4H > κ ≥ 4 -2H.

1 with θ = 0, γ = 2 -2H is equivalent to H < 3/
Then assumption (A.2) 2 holds.

The following property provides a sufficient condition for (A.2) p in spectral terms, which does not require differentiability of the variogram. Property 4.3 Let X be a Gaussian process having stationary increments and the spectral representation (see for instance Cramèr and Leadbetter, 1967)

X t = R e itξ -1 f 1/2 (ξ) W (dξ), for all t ∈ R, (4.16)
where W (dx) = W (-dx) is a complex-valued Gaussian white noise with zero mean and variance

E |W (dx)| 2 =
dx and f is a non-negative even function called the spectral density of X such that

R 1 ∧ |ξ| 2 f (ξ) dξ < ∞. (4.17)
Moreover, assume that f is differentiable on (K, ∞) and

f (ξ) ∼ c ξ -2H-1 (ξ → ∞), |f ′ (ξ)| ≤ C ξ -2H-2 (ξ > K) (4.18)
for some constants c, C, K > 0. Then X satisfies assumption (A.2) 1 for 0 < H < 3/4 and assumption (A.2) 2 for 0 < H < 1.

Limit theorems

Before establishing limit theorems for the statistics R p,n for Gaussian processes, recall that λ is given in (1.9) and with ρ p (H) in (6.31) one has

1 0 Λ p (H(t))dt = 1 0 λ(ρ p (H(t)))dt = 1 0 E ∆ p 0 B H(t) + ∆ p 1 B H(t) ∆ p 0 B H(t) + ∆ p 1 B H(t)
dt.

Straightforward computations show that Assumptions (A.1) and (A.2) p imply Assumption (A) with The following Theorem 4.2 is also established in [START_REF] Bardet | A central limit theorem for triangular arrays of nonlinear functionals of Gaussian vectors[END_REF]. Its proof (see a sketch of this proof in Section 6) uses a general central limit theorem for Gaussian subordinated nonstationary triangular arrays (see Theorem 1 in Section 6). Note that the Hermite rank of ψ(

A (t) (δ) = δ -H(t) , Y (t) = c(t)
x 1 , x 2 ) = |x 1 + x 2 |/(|x 1 | + |x 2 |) is 2
and this explains the difference between the cases p = 1 and p = 2 in Theorem 4.2: in the first case, the inequalities in (6.12) for (Y n (k)) as defined in (6.9)-(6.10) hold only if sup t∈[0,1] H(t) < 3/4, while in the latter case these inequalities hold for 0 < sup t∈[0,1] H(t) < 1. A similar fact is true also for the estimators based on generalized quadratic variations, see Istas and Lang (1997), [START_REF] Coeurjolly | Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths[END_REF]. Theorem 4.2 Let X be a Gaussian process satisfying assumptions (A.1) ′ and (A.2) p , with θ = 0. Moreover, assume additionally sup

t∈[0,1] H(t) < 3/4 if p = 1. Then, for p = 1, 2, √ n R p,n - 1 0 Λ p (H(t))dt D -→ n→∞ N 0, 1 0 Σ p (H(τ ))dτ , (4.21)
with Λ p (H) and Σ p (H) given in (1.8) and (1.12), respectively.

The following proposition shows that the previous theorems are satisfied when smooth multiplicative and additive trends are considered.

Proposition 4.1 Let Z = (Z t = α(t)X t + β(t), t ∈ [0, 1]), where X = (X t , t ∈ [0, 1]
) is a zero mean Gaussian process and α, β are deterministic continuous functions on [0, 1] with inf t∈[0,1] α(t) > 0. The particular case of Gaussian processes having stationary increments can also be studied: Corollary 4.2 Assume that X is a Gaussian process having stationary increments and there exist c > 0, C > 0 and 0 < H < 1 such that at least one of the two following conditions (a), (b) hold:

(a) variogram V (t) = c t 2H 1 + o(t 1/2 ) for t → 0 and |V (2p) (t)| ≤ Ct 2H-2p for all t ∈ (0, 1];
(b) spectral density f satisfies (4.17), (4.18) and

f (ξ) = cξ -2H-1 1 + o(ξ -1/2 ) (ξ → ∞). Then: √ n R p,n -Λ p (H) D -→ n→∞ N 0, Σ p (H) if p = 1, 0 < H < 3/4, p = 2, 0 < H < 1. (4.22)
Moreover, with the expression of s 2 2 (H) given in Section 6,

√ n Λ -1 2 (R 2,n ) -H D -→ n→∞ N 0, s 2 2 (H) . (4.23)
Therefore, Another interesting particular case of Theorem 4.2 leads to a punctual estimator of the function H(t) from a localization of the statistic R 2,n . For t 0 ∈ (0, 1) and α ∈ (0, 1), define

H n = Λ -1 2 (R
R 2,n α (t 0 ) := 1 2n α [nt0+n α ] k=[nt0-n α ] ∆ 2,n k X + ∆ 2,n k+1 X |∆ 2,n k X| + |∆ 2,n k+1 X| .
This estimator is studied in Bardet

Examples

Below we provide some concrete examples of Gaussian processes which admit a fBm as the tangent process.

For some examples the hypotheses of Theorems 4.1-4.2 and the subsequent corollaries are satisfied. For other examples, the verification of our hypotheses (in particular, of the crucial covariance bound (A.2) p ) remains an open problem and will be discussed elsewhere.

Example 4.1 Fractional Brownian motion (fBm).

As noted above, a fBm X = B H satisfies (A.1) ′ as well as (A.2) 1 (for 0 < H < 3/4 if θ = 0 and 0 < H < 7/8 if 0 < θ < 2 -2H with θ arbitrary close to 2 -2H and therefore γ < 2 -2H + θ arbitrary close to 4 -4H may satisfy γ > 1/2) and (A.2) 2 (for 0 < H < 1), with H(t) ≡ H, c(t) ≡ c. Therefore, for fBm both Theorems 4.1 (the almost sure convergence, satisfied for 0 < H < 7/8 when p = 1 and for 0 < H < 1 when p = 2) and 4.2 (the central limit theorem, satisfied for 0 < H < 3/4 when p = 1 and for 0 < H < 1 when p = 2)) apply. Obviously, a fBm also satisfies the conditions of Corollary 4.2. Thus, the rate of convergence of the estimator Λ -1 2 (R 2,n ) =: H n of H is √ n. But in such a case the self-similarity property of fBm allows to use in this case asymptotically efficient Whittle or maximum likelihood estimators (see [START_REF] Fox | Large-sample properties of parameter estimates for strongly dependent Gaussian time series[END_REF]Taqqu, 1987, or Dahlhaus, 1989). However, for a fBm with a continuously differentiable multiplicative and additive trends, which leads to a semi-parametric context, the convergence rate of H n is still √ n while parametric estimators cannot be applied.

Example 4.2 Multiscale fractional Brownian motion (see [START_REF] Bardet | Definition, properties and wavelet analysis of multiscale fractional Brownian motion[END_REF]) defined as follows: for ℓ ∈ N * , a (M ℓ )-multiscale fractional Brownian motion X = (X t , t ∈ R) ((M ℓ )-fBm for short) is a Gaussian process having stationary increments and a spectral density f such that

f (ξ) = σ 2 j |ξ| 2Hj +1 1(ω j ≤ |ξ| < ω j+1 ) for all ξ ∈ R (4.24)
with

ω 0 := 0 < ω 1 < • • • < ω ℓ < ω ℓ+1 := ∞, σ i > 0 and H i ∈ R for i ∈ {0, • • • , ℓ} with H 0 < 1 and H ℓ > 0. Therefore condition (4.18) of Property 4.3 is satisfied, with K = ω ℓ and H = H ℓ . Moreover, the condition f (ξ) = cξ -2H-1 1 + o(ξ -1/2 ) (ξ → ∞) required in Corollary 4.2 is also checked with H = H ℓ .
Consequently, the same conclusions as in the previous example apply for this process as well, in the respective regions determined by the parameter H ℓ at high frequencies x > ω ℓ alone. The same result is also obtained for a more general process defined by f (ξ) = cξ -2H-1 for |ξ| ≥ ω and condition (4.17) is only required elsewhere. Once again, such conclusions hold also in case of continuously differentiable multiplicative and additive trends.

Example 4.3 Multifractional Brownian motion (mBm) (see [START_REF] Ayache | Regularity and identification of generalized multifractional Gaussian processes[END_REF]. A mBm X = (X t , t ∈ [0, 1]) is a Gaussian process defined by

X t = B H(t) (t) = g(H(t)) R e itx -1 |x| H(t)+1/2 W (dx), (4.25)
where W (dx) is the same as in (4.16), H(t) is a (continuous) function on [0, 1] taking values in (0, 1) and finally, g(H(t)) is a normalization such that EX 2 t = 1. It is well-known that a mBm is locally asymptotically self-similar at each point t ∈ (0, 1) having a fBm B H(t) as its tangent process at t (see [START_REF] Benassi | Gaussian processes and pseudodifferential elliptic operators[END_REF]. This example is studied in more detail in [START_REF] Bardet | Nonparametric estimation of the local Hurst function of multifractional processes[END_REF].

Example 4.4 Time-varying fractionally integrated processes. [START_REF] Philippe | Invariance principle for a class of non stationary processes with long memory[END_REF][START_REF] Philippe | Time-varying fractionally integrated processes with nonstationary long memory[END_REF] introduced two classes of mutually inverse time-varying fractionally integrated filters with discrete time and studied longmemory properties of the corresponding filtered white noise processes. [START_REF] Surgailis | Nonhomogeneous fractional integration and multifractional processes[END_REF] extended these filters to continuous time and defined "multifractional" Gaussian processes (X t , t ≥ 0) and (Y t , t ≥ 0) as follows

X t = R t 0 1 Γ(H(τ ) -.5) (τ -s) H(τ )-1.5 +
e A-(s,τ ) dτ dB(s), (4.26)

Y t = R 1 Γ(H(s) + .5) (t -s) H(s)-.5 + e -A+(s,t) -(-s) H(s)-.5 +
e -A+(s,0) dB(s), (4.27) where

s α + := s α 1(s > 0), B is a Brownian motion, A -(s, t) := t s H(u) -H(t) t -u du, A + (s, t) := t s H(s) -H(v) v -s dv (s < t)
and where H(t), t ∈ R is a general function taking values in (0, ∞) and satisfying some weak additional conditions. Surgailis (2008) studied small and large scale limits of (X t ) and (Y t ) and showed that these processes resemble a fBm with Hurst parameter H = H(t) at each point t ∈ R (i.e., admit a fBm as a tangent process) similarly to the mBm in the previous example. The last paper also argues that these processes present a more natural generalization of fBm than the mBm and have nicer dependence properties of increments. We expect that the assumptions (A.1), (A.1) ′ , (A.2) p can be verified for (4.26), (4.27); however, this question requires further work.

Processes with independent increments

In this section, we assume that X = (X t , t ≥ 0) is a (homogeneous) Lévy process, with a.s. right continuous trajectories, X 0 = 0. It is well-known that if the generating triplet of X satisfies certain conditions (in particularly, if the Lévy measure ν behaves regularly at the origin with index α ∈ (0, 2)), then X has a tangent process Y which is α-stable Lévy process. A natural question is to estimate the parameter α with the help of the introduced statistics R p,n . Unfortunately, the limit of these statistics as defined in (1.5) through the tangent process depends also on the skewness parameter β [-1, 1] of the α-stable tangent process Y and so this limit cannot be used for determining of α if β is unknown.

In order to avoid this difficulty, we shall slightly modified our ratio statistic, as follow. Observe first that the second differences ∆ 2,n k X of Lévy process have a symmetric distribution (in contrast to the first differences ∆ 1,n k X which are not necessary symmetric). For notational simplicity we shall assume in this section that n is even. The modified statistic

R2,n := 1 n/2 -1 (n-4)/2 k=0 ψ ∆ 2,n 2k X, ∆ 2,n 2k+2 X , ψ(x, y) := |x + y| |x| + |y|
is written in terms of "disjoint" (independent) second order increments (∆ 2,n 2k X, ∆ 2,n 2k+2 X) having a symmetric joint distribution. Instead of extending general result of Proposition 2.2 to R2,n , we shall directly obtain its convergence under suitable assumptions on X. Note first

E R2,n = Eψ X (2) 1/n -X (1) 1/n , X (4) 1/n -X (3) 1/n , (5.1)
where

X (i) , i = 1, • • • , 4 are independent copies of X. Note that 1/2 ≤ E R2,n ≤ 1 since Eψ X (2) 1/n -X (1) 1/n , X (4)
1/n -X

(3) -→ n→∞ Λ.

1/n ≥ P(X (2) 1/n -X (1) 1/n ≥ 0, X (4) 1/n -X (3) 1/n ≥ 0) + P(X (2) 1/n -X (1) 1/n < 0, X (4) 1/n -X (3) 1/n < 0) ≥ P 2 (X (2) 1/n -X (1) 1/n ≥ 0) + P 2 (X (2) 1/n -X (1) 1/n < 0) ≥ 1/2.
(5.3)

Proof. Write R2,n = E R2,n + (n/2 -1) -1 Q n ,
where Q n is a sum of centered 1-dependent r.v.'s which are bounded by 1 in absolute value. Therefore E((n/2 -1)

-1 Q n ) 4 = O(n -2
) and the a.s. convergence (n/2 -1) -1 Q n → 0 follows by the Chebyshev inequality.

Next we discuss conditions on X for the convergence in (5.2). Recall that the distribution of X t is infinitely divisible and its characteristic function is given by

Ee iθXt = exp t iγθ - 1 2 a 2 θ 2 + R e iuθ -1 -iuθ1(|u| ≤ 1) ν(du) , θ ∈ R, (5.4)
where γ ∈ R, a ≥ 0 and ν is a measure on R such that R min(u 2 , 1)ν(du) < ∞. The triplet (a, γ, ν) is called the generating triplet of X (Sato (1999)). Let X (i) , i = 1, 2 be independent copies of X. Note W t := X

(1) t -X

(2) t is a Lévy process having the characteristic function

Ee iθWt = exp t -a 2 θ 2 + 2 ∞ 0 Re 1 -e iuθ dK(u) , θ ∈ R, (5.5)
where

K(u) := ν((-∞, -u] ∪ [u, ∞))
is monotone nonincreasing on (0, ∞). Introduce the following condition: there exist 0 < α ≤ 2 and c > 0 such that

K(u) ∼ c u α , u ↓ 0. (5.6)
It is clear that if such number α exists then α := inf r ≥ 0 : |x|≤1 |x| r ν(dx) < ∞ is the so-called fractional order or the Blumenthal-Getoor index of the Lévy process X.

Let Z α be a standard α-stable r.v. with characteristic function Ee iθZα = e -|θ| α and Z

α , i = 1, 2, 3 be independent copies of Z α . Proposition 5.2 Assume either a > 0 or else, a = 0 and condition (5.6) with 0 < α ≤ 2 and c > 0. Then t -1/α (X

Λ ≡ Λ(α) := Eψ Z (1) α , Z (2)
α .

Moreover, with σ2 (α) := 2var ψ Z (1) α , Z (2) α

+ 4cov ψ Z (1) α , Z (2) α , ψ Z (2) α , Z (3) α , √ n( R2,n -E R2,n ) D -→ n→∞ N (0, σ2 (α)).
(5.7)

Proof. Relation t -1/α W t = t -1/α (X (1) t -X (2) t )
D -→ t→0 c Z α is an easy consequence of the assumptions of the proposition and the general criterion of weak convergence of infinitely divisible distributions in Sato (1991, Theorem 8.7). It implies (5.2) by the fact that ψ is a.e. continuous on R 2 . Since R2,n is a sum of 1-dependent stationary and bounded r.v.'s, the central limit theorem in (5.7) follows from convergence of the variance:

n var( R2,n ) → σ2 (α);

(5.8) see e.g. [START_REF] Berk | A central limit theorem for m-dependent random variables with unbounded m[END_REF]. Rewrite R2,n = (n/2 -1) -1 (n-4)/2 k=0 ηn (k), ηn (k) := ψ ∆ 2,n 2k X, ∆ 2,n 2k+2 X . We have

n var( R2,n ) = n n/2 -1 var(η n (0)) + 2n(n/2 -2) (n/2 -1) 2 cov(η n (0), ηn (1)),
where var(η n (0)) -→ n→∞ var ψ Z

(1)

α , Z (2) α
, cov(η n (0), ηn (1)) -→

n→∞ cov ψ Z (1) α , Z (2) α
, ψ Z

α , Z

(3) α similarly as in the proof of (5.2) above. This proves (5.8) and the proposition. The graph of Λ(α) is given in Figure 4. Note that Λ(2) = Λ 1 (1/2) ≃ 0.72: this is the case of Brownian motion. In order to evaluate the decay rate of the bias E R2,n -Λ(α) we need a uniform convergence rate in Lemma 5.1, below, for

F n -G α ∞ := sup x∈R |F n (x) -G α (x)|, F n (x) := P(n 1/α W 1/n ≤ x), G α (x) := P( Zα ≤ x),
where Zα := cZ α is the limiting α-stable r.v. in Proposition 5.2 and (W t , t ≥ 0) is the symmetric Levy process with characteristic function as in (5.5). The proof of Lemma 5.1 is given in Annexe. Lemma 5.1 (i) Let a = 0 and K satisfy (5.6)

. Denote K 1 (u) := K(u) -cu -α , |K 1 |(u) := ∞ u |dK 1 (v)|, the variation of K 1 on [u, ∞)
. Moreover, assume that there exist some constants β, δ > 0 such that

|K 1 |(u) = O(u -(α-β)+ ) (u → 0), |K 1 |(u) = O(u -δ ) (u → ∞), (5.9)
where x + := max(0, x). Then

F n -G α ∞ =      O(n -β/α ), if β < α, O(n -1 log n), if β = α, O(n -1 ), if β > α.
(5.10)

(ii) Let a > 0 and K satisfy

K(u) = O(u -α ) (u → 0), K(u) = O(u -δ ) (u → ∞), (5.11)
for some 0 ≤ α < 2, δ > 0. Then

F n -G α ∞ = O(n -1+α/2 ), if α > 0, O(n -1 log n), if α = 0.
(5.12)

Proposition 5.3 Assume either a > 0 or else a = 0 and condition (5.6). Then for any α ∈ (0, 2]

|E R2,n -Λ(α)| ≤ 2C F n -G α ∞ , C := ∞ 0 (1 + z) -2 dz < ∞. (5.13)
Proof. Let ψ(x, y) := |x -y|/(x + y), x, y > 0, and let F n , G α be the same as in Lemma 5.1. Similarly as in Vaičiulis (2009, proof of Th. 1), write

E R2,n -Λ(α) = 2 ∞ 0 ∞ 0 ψ(x, y)(dF n (x)dF n (y) -dG α (x)dG α (y)) = 2(W 1 + W 2 ),
where

W 1 := ∞ 0 ∞ 0 ψ(x, y)dF n (x)(dF n (y) -dG α (y)), W 2 := ∞ 0 ∞ 0 ψ(x, y)dG α (y)(dF n (x) -dG α (x)
). Integrating by parts yields

|W 1 | = 2 ∞ 0 |x|dF n (x) ∞ 0 |F n (y) -G α (y)| dy (x + y) 2 ≤ 2 F n -G α ∞ ∞ 0 |x|dF n (x) ∞ 0 dy (x + y) 2 = C F n -G α ∞ since ∞ 0 dF n (x) = 1/2.
A similar estimate holds for W 2 . This proves (5.13). Propositions 5.2, 5.3, and Lemma 5.1, together with the Delta-method, yield the following corollary.

Corollary 5.1 Let a and K satisfy either the assumptions of Lemma 5.1(i) with β > α/2, or the assumptions of Lemma 5.1(ii)

. Then √ n( R2,n -Λ(α)) D -→ n→∞ N (0, σ2 (α))
.

Moreover, if we define α

n := Λ-1 ( R2,n ), then √ n α n -α D -→ n→∞ N (0, s2 (α))
,

where s2 (α) := ∂ Λ ∂α (α) -2 σ2 (α), 0 < α ≤ 2.
There exist very few papers concerning estimation of α in such a semiparametric frame. Nonparametric estimation of parameters of Lévy processes based on the empirical characteristic function has recently been considered in [START_REF] Neumann | Nonparametric estimation for Lévy processes from low-frequency observations[END_REF] and [START_REF] Gugushvili | Nonparametric estimation of the characteristic triplet of a discretely observed Lévy process[END_REF], but the convergence rates there are (log n) κ with κ > 0. Ait Sahalia and Jacod (2009) have proposed an estimator of the degree of activity of jumps (which is identical to the fractional order in the case of a Lévy process) in a general semimartingale framework using small increments of high frequency data. However from the generality of their model, the convergence rate of the estimator is not rate efficient (in fact smaller than n 1/5 ). A recent paper of [START_REF] Belomestny | Spectral estimation of the fractional order of a Lévy process[END_REF] provides an efficient data-driven procedure to estimate α using a spectral approach but in a different semiparametric frame from ours. Thus, Corollary 5.1 appears as a new and interesting result since the estimator α n follows a √ n-central limit theorem.

Annexe : proofs

Proof of Lemma 3.1

Let δ 2 := max i=1,2 Eξ 2 i . If δ 2 ≥ 1/2, then (3.2) holds since the l.h.s. of (3.2) does not exceed 1. Let δ 2 < 1/2 in the sequel. Write U := ψ(Z 1 + ξ 1 , Z 2 + ξ 2 ) -ψ(Z 1 , Z 2 ) = U δ + U c δ , U δ := U 1(A δ ) = (ψ(Z 1 + ξ 1 , Z 2 + ξ 2 ) -ψ(Z 1 , Z 2 ))1(A δ ), U c δ := U 1(A c δ ) = (ψ(Z 1 + ξ 1 , Z 2 + ξ 2 ) -ψ(Z 1 , Z 2 ))1(A c δ ),
where 1(A δ ) is the indicator of the event

A δ := {|Z 1 | > δ 2/3 , |Z 2 | > δ 2/3 , |ξ 1 | < δ 2/3 /2, |ξ 2 | < δ 2/3 /2}, and
1(A c δ ) = 1 -1(A δ ) is the indicator of the complementary event A c . Clearly, E|U c δ | ≤ 2(P(|Z 1 | < δ 2/3 ) + P(|Z 2 | < δ 2/3 ) + P(|ξ 1 | ≥ δ 2/3 /2) + P(|ξ 2 | ≥ δ 2/3 /2) ≤ 4 √ 2π δ 2/3 + 2 Eξ 2 δ 4/3 ≤ 4 δ 2/3 .
It remains to estimate E|U δ |. By the mean value theorem,

|U δ | ≤ |ξ 1 | sup D |ψ x1 | + |ξ 2 | sup D |ψ x2 | 1(A δ ),
where sup

D |ψ x1 | := sup{|∂ψ(x 1 , x 2 )/∂x 1 | : |x i -Z i | ≤ |ξ i |, i = 1, 2}, sup D |ψ x2 | := sup{|∂ψ(x 1 , x 2 )/∂x 2 | : |x i -Z i | ≤ |ξ i |, i = 1, 2}.
Therefore

E|U δ | ≤ E 1/2 (ξ 2 1 )E 1/2 (sup D |ψ x1 |) 2 1(A δ ) + E 1/2 (ξ 2 2 )E 1/2 (sup D |ψ x2 |) 2 1(A δ ) ≤ δ E 1/2 (sup D |ψ x1 |) 2 1(A δ ) + E 1/2 (sup D |ψ x2 |) 2 1(A δ ) .
Next,

|ψ xi (x 1 , x 2 )| = |sgn(x 1 + x 2 )(|x 1 | + |x 2 |) -(|x 1 + x 2 |)sgn(x i )| (|x 1 | + |x 2 |) 2 ≤ 2 |x 1 | + |x 2 | . Therefore sup D |ψ xi | 2 1(A δ ) ≤ 4 sup (|x 1 | + |x 2 |) -2 : |x i -Z i | ≤ δ 2/3 /2, |Z i | > δ 2/3 , i = 1, 2 ≤ 16 (|Z 1 | + |Z 2 |) -2 1(|Z i | > δ 2/3 , i = 1, 2), implying E (sup D |ψ xi |) 2 1(A δ ) ≤ 16 E 1 (|Z 1 | + |Z 2 |) 2 ; |Z i | > δ 2/3 , i = 1, 2 ≤ C(δ),
where

C(δ) ≤ 16 2π {x 2 1 +x 2 2 >δ 4/3 } 1 x 2 1 + x 2 2 e -(x 2 1 +x 2 2 )/2 dx 1 dx 2 ≤ 16 ∞ δ 4/3
r -1 e -r 2 /2 dr ≤ 16 (1 + (4/3)| log δ|).

Hence E|U δ | ≤ 8 δ(1 + (4/3) | log δ|) 1/2 , E|U c δ | ≤ 4 δ 2/3 . It remains to use x(1 + (4/3) | log x|) 1/2 ≤ 2x 2/3 for all 0 < x ≤ 1.
.

Proof of Property 4.1

We use the following identity: for any reals x 1 , • • • , x j ,

x 1 x j = 1 2 j k=1 x k 2 + j-1 k=2 x k 2 - j-1 k=1 x k 2 - j k=2 x k 2 .
In particular,

E ∆ 1,n [nt] X∆ 1,n j+[nt] X = 1 2 E X j+1+[nt] -X [nt] 2 + E X j-1+[nt * ] -X [nt * ] 2 -E X j+[nt] -X [nt] 2 -E X j+[nt * ] -X [nt * ] 2 ,
where t * := t + (1/n) (so that [nt * ] = [nt] + 1). Then, using (A.1) and the notation u n for a sequence tending to 0 as n → ∞ uniformly in t and all |j| < J, where J is a fixed number, we obtain

E ∆ 1,n [nt] X∆ 1,n j+[nt] X = 1 2 c(t) j+1 n 2H(t) (1 + u n ) + c(t * ) j-1 n 2H(t * ) (1 + u n ) -c(t) j n 2H(t) (1 + u n ) -c(t * ) j n 2H(t * ) (1 + u n ) = c(t) 2 j+1 n 2H(t) + j-1 n 2H(t) -2 j+1 n 2H(t) (1 + u n ) since c(t * ) -c(t) = u n and j n 2H(t * ) = j n 2H(t)
(1 + u n ) follows from (4.5). This proves (4.9) for p = 1. Relation (4.9) for p = 2 follows analogously. Relation

Proof of Property 4.2

With condition V (t) ∼ ct 2H (t → 0) in mind, inequality (4.8) reduces to

V k+1 n + V k-1 n -2V k n ≤ Cn -2H+θ k -γ (p = 1, 2 ≤ k ≤ n), (6.1) V k+2 n -4V k+1 n + 6V k n -4V k-1 n + V k-2 n ≤ Cn -2H+θ k -γ (p = 2, 4 ≤ k ≤ n). (6.2)
The left hand side of (6.1) can be written and estimated as

1/n 0 1/n 0 V ′′ (t -s + (k/n))dtds ≤ C 1/n 0 1/n 0 |t -s + (k/n)| -κ dtds ≤ Cn κ-2 1 0 1 0 |t -s + k| -κ dtds ≤ Cn κ-2 k -κ = Cn θ-2H k -γ
where γ = κ > 1/2 and θ = κ + 2H -2 ∈ [0, γ/2) since κ < 4 -4H. This proves part (i). Part (ii) follows similarly, by writing the left hand side of (6.2) as

1/n 0 • • • 1/n 0 V (4) (t -s + u -v + (k/n))dtdsdudv ≤ C 1/n 0 • • • 1/n 0 |t -s + u -v + (k/n)| -κ dtdsdudv ≤ Cn κ-4 k -κ = Cn θ-2H k -γ ,
where γ = κ > 1 and θ = κ + 2H -4 ∈ [0, κ/2) since κ < 8 -4H. Property 4.2 is proved.

Proof of Property 4.3

From (4.16) we have

E ∆ p,n 0 X ∆ p,n j X = R e i(x/n) -1 2p e ix(j/n) f (x)dx = 2 1+p ∞ 0 (1 -cos(x/n)) p cos(xj/n)f (x)dx = 2 1+p (I 1 + I 2 ),
with

|I 1 | = K 0 (1 -cos(x/n)) p cos(xj/n)f (x)dx = C K 0 (x/n) 2p f (x)dx = Cn -2p K 0 x 2p f (x)dx ≤ Cn -2p ; (6.3)
and

I 2 = n ∞ K/n (1 -cos(x)) p cos(xj)f (nx)dx = (n/j) ∞ K/n (1 -cos(x)) p f (nx)d sin(xj) = -(n/j) ∞ K/n sin(xj) ((1 -cos(x)) p f (nx)) ′ x dx + O(n -2p ) (6.4) since |(n/j)f (K)(1 -cos(K/n)) p sin(jK/n)| ≤ C(n/j)(K/n) 2p |jK/n| ≤ Cn -2p
for any K > 0 fixed.

Let p = 1. The last integral can be rewritten as

∞ K/n sin(xj) ((1 -cos(x))f (nx)) ′ x dx = Ĩ1 + Ĩ2 , where | Ĩ1 | ≤ ∞ K/n | sin(xj) sin(x)f (nx)|dx ≤ C ∞ K/n | sin(xj) sin(x)|(nx) -2H-1 dx ≤ 3 q=1 I 1q ,
where we used the fact that f (x) ≤ Cx -2H-1 (x ≥ K); see condition (4.18), and where

|I 11 | = 1/j 0 | sin(xj) sin(x)|(nx) -2H-1 dx ≤ Cj 0 x 2 (nx) -1-2H dx ≤ Cj 2H-1 n -1-2H , |I 12 | = 1 1/j | sin(xj) sin(x)|(nx) -2H-1 dx ≤ C 1 1/j x(nx) -1-2H dx ≤ Cj 2H-1 n -1-2H , |I 13 | = ∞ 1 | sin(xj) sin(x)|(nx) -2H-1 dx ≤ C ∞ 1 (nx) -1-2H dx ≤ Cn -1-2H .
Similarly, using (4.18),

| Ĩ2 | ≤ n ∞ K/n | sin(xj)(1 -cos(x))f ′ (nx)|dx ≤ C ∞ K/n | sin(xj)(1 -cos(x)|(nx) -2H-2 dx ≤ 3 q=1 I 2q ,
where

I 21 = n 1/j 0 | sin(xj)(1 -cos(x))|(nx) -2H-2 dx ≤ Cnj 1/j 0 x 3 (nx) -2-2H dx ≤ Cj 2H-1 n -1-2H , I 22 = n 1 1/j | sin(xj)(1 -cos(x))|(nx) -2H-2 dx ≤ Cn 1 1/j x 2 (nx) -2-2H dx ≤ Cj 2H-1 n -1-2H , I 23 = n ∞ 1 | sin(xj)(1 -cos(x))|(nx) -2H-2 dx ≤ Cn ∞ 1 (nx) -2-2H dx ≤ Cn -1-2H .
We finally obtain, for 1 ≤ j ≤ n,

E ∆ 1,n 0 X ∆ 1,n j X ≤ C(n/j)j 2H-1 n -1-2H + O(n -2 ) ≤ Cn -2H j 2H-2
implying (for 0 < H < 3/4) (A.2) 1 with θ = 0 and κ = 2 -2H > 1/2. For p = 2, the estimation of the integral in (6.4) is completely similar, resulting in the bound

E ∆ 2,n 0 X ∆ 2,n j X ≤ C(n/j)n -1-2H + O(n -4 ) ≤ Cn -2H j -1
for any 0 < H < 1, or (A.2) 2 with θ = 0 and κ = 1 > 1/2.

Sketch of the proof of Theorem 4.1

The proof of Theorem 4.1 is based on the moment inequality in Lemma 1, below, which extends a similar inequality in (Taqqu, 1977, Lemma 4.5) to vector-valued nonstationary Gaussian processes. The proof of Lemma 1 uses the diagram formula and is given in [START_REF] Bardet | A central limit theorem for triangular arrays of nonlinear functionals of Gaussian vectors[END_REF]. To formulate this lemma, we need the following definitions. Let X be a standard Gaussian vector in R ν (ν ≥ 1) and let L 2 (X) denote the Hilbert space of measurable functions f : R ν → R satisfying

f 2 := E(f (X)) 2 < ∞. Let L 2 0 (X) = {f ∈ L 2 (X) : Ef (X) = 0}. Let (X 1 , • • • , X N ) be a collection of standardized Gaussian vectors X t = (X (1)
t , • • • , X (ν)
t ) ∈ R ν having a joint Gaussian distribution in R νN . Let ε ∈ [0, 1] be a fixed number. Following [START_REF] Taqqu | Law of the iterated logarithm for sums of non-linear functions of Gaussian variables that exhibit a long range dependence[END_REF]

, we call (X 1 , • • • , X N ) ε-standard if |EX (u) t X (v) s | ≤ ε for any t = s, 1 ≤ t, s ≤ N and any 1 ≤ u, v ≤ ν. Finally, ′ denotes the sum over all distinct integers 1 ≤ t 1 , • • • , t p ≤ N, t i = t j (i = j). Lemma 1 Let (X 1 , • • • , X N ) be ε-standard Gaussian vector, X t = (X (1)
t , • • • , X (ν) t ) ∈ R ν (ν ≥ 1), and let G j,t,N ∈ L 2 (X), 1 ≤ j ≤ p (p ≥ 2), 1 ≤ t ≤ N . For given integers m, N ≥ 1, define Q N := max 1≤t≤N 1≤s≤N,s =t max 1≤u,v≤ν |EX (u) t X (v) s | m . (6.5)
Assume that for some integer 0 ≤ α ≤ p, the functions G 1,t,N , • • • , G α,t,N have a Hermite rank at least equal to m for any N ≥ 1, 1 ≤ t ≤ N , and that ε <

1 νp-1 . Then ′ E|G 1,t1,N (X t1 ) • • • G p,tp,N (X tp )| ≤ C(ε, p, m, α, ν)KN p-α 2 Q α 2 N ,
where the constant C(ε, p, m, α, ν) depends on ε, p, m, α, ν only, and K := ηn (k), where ηn (k) := η n (k) -Eη n (k) and η n (k) := ψ(∆ p,n k X, ∆ p,n k+1 X), ψ(x, y) = |x + y|/(|x| + |y|) are nonlinear functions of Gaussian vectors (∆ p,n k X, ∆ p,n k+1 X) ∈ R 2 having the Hermite rank 2; however, these vectors are not ε-standard and therefore Lemma 1 cannot be directly applied to estimate the l.h.s. of (6.7) (with p = 1, • • • , 4, ν = 2). To this end, we first need to "decimate" the sum Rp,n , as follows. (A similar "trick" was used in [START_REF] Csörgő | The empirical process of a short-range dependent stationary sequence under Gaussian subordination[END_REF].) Let ℓ = [n θ/γ ] be the sequence of integers increasing to ∞ (at a rate o(n 1/2 ) by condition θ < γ/2) and write

Rp,n = ℓ-1 j=0 Rp,n ℓ (j) + o(1), Rp,n ℓ (j) := 1 n -1 [(n-2-j)/ℓ] k=0 ηn (kℓ + j). Then E Rp,n 4 ≤ ℓ 4 max 0≤j<ℓ E Rp,n ℓ (j) 4 .
Write η n (k) as a (bounded) function in standardized Gaussian variables:

η n (k) = f k,n (Y n (k)) , (6.8)
where

Y n (k) = (Y (1)
n (k), Y (2)
n (k)) ∈ R 2 , Y (1) n (k) := ∆ p,n k X σ p,n (k) , (6.9)
Y (2) n (k) := - ∆ p,n k X σ p,n (k) ρ p,n (k) 1 -ρ 2 p,n (k) + ∆ p,n k+1 X σ p,n (k + 1) 1 1 -ρ 2 p,n (k) , (6.10)
and f k,n x (1) , x (2) := ψ x (1) , σ p,n (k + 1)

σ p,n (k) ρ p,n (k)x (1) + 1 -ρ 2 p,n (k)x (2) , (6.11)
where σ 2 p,n (k), ρ p,n (k) are defined in (4.1). Then, for each k, Y n (k) := (Y

n (k), Y (1)
n (k)) has a standard Gaussian distribution in R 2 and ηn (k

) = f k,n (Y n (k))-Ef k,n (Y n (k)). Moreover, the vector (Y n (kℓ+j), k = 0, 1, • • • , [(n -2 -j)/ℓ]) ∈ R 2([(n-2-j)/ℓ]+1
) is ε-standard provided ℓ is large enough. Now Lemma 1 can be used and it implies the bound (6.7) using Assumptions (A.1) and (A.2) p . The details of this proof can be found in [START_REF] Bardet | A central limit theorem for triangular arrays of nonlinear functionals of Gaussian vectors[END_REF].

Sketch of the proof of Theorem 4.2

The proof of Theorem 4.1 uses the following central limit theorem for Gaussian subordinated multidimensional triangular arrays. Theorem 1 is proved in [START_REF] Bardet | A central limit theorem for triangular arrays of nonlinear functionals of Gaussian vectors[END_REF]. It extends the earlier results in [START_REF] Breuer | Central limit theorems for nonlinear functionals of Gaussian fields[END_REF] and [START_REF] Arcones | Limit theorems for nonlinear functionals of a stationary Gaussian sequence of vectors[END_REF]. Below, similarly as in Lemma 1, X ∈ R ν designates a standard Gaussian vector.

Theorem 1 Let (Y n (k)) 1≤k≤n,n∈N be a triangular array of standardized Gaussian vectors with values in

R ν , Y n (k) = (Y (1) n (k), • • • , Y (ν) n (k)), EY (p) n (k) = 0, EY (p) n (k)Y (q)
n (k) = δ pq . For a given integer m ≥ 1, introduce the following assumption: there exists a function ρ : N → R such that for any 1 ≤ p, q ≤ ν,

∀(j, k) ∈ {1, • • • , n} 2 , EY (p) n (j)Y (q) n (k) ≤ |ρ(j -k)| with j∈Z |ρ(j)| m < ∞. (6.12)
Moreover, assume that for any τ ∈ [0, 1] and any J ∈ N * ,

(Y n ([nτ ] + j)) -J≤j≤J D -→ n→∞ (W τ (j)) -J≤j≤J , (6.13)
where (W τ (j)) j∈Z is a stationary Gaussian process taking values in R ν and depending on parameter τ ∈ (0, 1). Let fk,n ∈ L 2 0 (X) (n ≥ 1, 1 ≤ k ≤ n) be a triangular array of functions all having Hermite rank at least m. Assume that there exists a L 2 0 (X)-valued continuous function φτ , τ ∈ [0, 1], such that The proof of (6.16) uses Assumption (A.1)' or (4.10) and the easy fact that for Gaussian vectors (Z

sup τ ∈[0,1] f[τn],n -φτ 2 = sup τ ∈[0,1] E( f[τn],n (X) -φτ (X)) 2 -→ n→∞ 0. (6.14) Then, with σ 2 = 1 0 dτ j∈Z E φτ (W τ (0)) φτ (W τ (j)) < ∞, n -1/2 n k=1 fk,n (Y n (k)) D -→ n→∞ N (0, σ 2 ). ( 6
n , Z

n ) ∈ R 2 , n ∈ N with zero mean EZ (i) n ≡ 0, i = 1, 2, n ∈ N and E(Z (1) 0 ) 2 = E(Z (2) 0 ) 2 = 1, |EZ (1) 0 Z (2) 0 | < 1 Eψ(Z (1) 0 Z (2) 0 ) -Eψ(Z (1) n , Z (2) n ) ≤ C 2 i,j=1 EZ (i) 0 Z (j) 0 -Z (i) n Z (j) n . (6.18) (2)
The proof of (6.17) is deduced from Theorem 1 with the sequence of standardized Gaussian vectors Y n (k) = (Y

n (k), Y (1)
n (k)) (ν = 2) given in (6.9)-(6.10) and the centered functions fk,n (x (1) , x (2) ) := f k,n (x (1) , x (2) ) -Ef k,n (Y n (k)), φτ (x (1) , x (2) ) := φ τ (x (1) , x (2) ) -Eφ τ (X) with f k,n : R 2 → R given in (6.11) and the (limit) function φ τ (x (1) , x (2) ) := ψ x (1) , ρ p (H(τ ))x (1) + 1ρ 2 p (H(τ ))x (2) .

Thanks to symmetry properties of these functions, it is clear that the Hermite rank of fk,n (for any k and n) and φτ (for any τ ∈ [0, 1]) is m = 2. Using Assumptions (A.1)' and (A.2) p (with θ = 0, γ > 1/2), one can show that the conditions of Theorem 1 are satisfied for the above fn,k , φτ and the limit process (W τ (j)) j∈Z in (6.13) is written in terms of increments of fBm (B H(τ ) (j)) j∈Z :

W τ (j) := ∆ p 1 B H(τ ) (j), -ρ p (H(τ ))∆ p 1 B H(τ ) (j) + ∆ p 1 B H(τ ) (j + 1) / 1 -ρ 2 p (H(τ ))
having standardized uncorrelated components. The details of this proof can be found in [START_REF] Bardet | A central limit theorem for triangular arrays of nonlinear functionals of Gaussian vectors[END_REF].

Proof of Proposition 4.1 (1cos(x)) p cos(xj) x -2H-3/2 dx < ∞. Therefore condition (4.10) is satisfied and Theorem 4.2 can be applied. Finally, the function H → Λ 2 (H) is a C 1 (0, 1) bijective function and from the Delta-method (see for instance Van der Vaart, 1998), the central limit theorem in (4.23) is shown.

Proof of Lemma 5.1 (i) We use the following inequality (see, e.g. Ibragimov and Linnik, 1971, Theorem 1.5.2):

F n -G α ∞ ≤ 1 2π R f n (θ) -g α (θ) θ dθ, (6.30)
where f n , g α are the characteristic functions of F n , G α , respectively. According to (5.6) and the definition of If v > 1 then integrating by parts and using (5.9),

|I 2 | ≤ 2|K 1 |(1/v) = O(v (α-β)+ ), |I 1 | = K 1 (1/v)Re(e i -1) - 1/v 0 K 1 (u) v sin(uv) du ≤ 2|K 1 (1/v)| + v 2 1/v 0 u|K 1 (u)|du = O(v (α-β)+ ) + v 2 (1/v) 2-(α-β)+ = O(v (α-β)+ ).
Next, if v ≤ 1, then similarly as above Therefore, for some constant C, |I n (θ)| ≤ C (θn 1/α ) (α-β)+ , θ > 1/n 1/α , (θn 1/α ) δ∧2 , θ ≤ 1/n 1/α . Moreover, g α (θ) = e -c1|θ| α for some c 1 > 0. Using these facts and (6.30) we obtain

|I 2 | ≤ 2|K 1 |(1/v) = O(v δ ),
F n -G α ∞ ≤ C 1/n 1/α 0 n -1 (θn 1/α ) δ∧2 dθ θ + + ∞ 1/n 1/α
n -1 (θn 1/α ) (α-β)+ e -c1|θ| α dθ θ ≤ Cn -1 + C 1 n 1-(α-β)+/α 1 + 1 1/n 1/α θ (α-β)+-1 dθ .

The above bound easily yields (5.10).

(ii) Follows similarly using (6.30) and the argument in (i) with K 1 replaced by K.

Computation of λ(r)

From the definition of λ(r) and the change of variables x 1 = a cos φ, x 2 = a sin φ, with |r| < 1, The function λ(r) is monotone increasing on [-1, 1]; λ(1) = 1, λ(-1) = 0. It is easy to check that ρ 1 (H) = 2 2H-1 -1, ρ 2 (H) = -3 2H + 2 2H+2 -7 8 -2 2H+1 (6.31) are monotone increasing functions; ρ 1 (1) = 1, ρ 2 (1) = 0 so that Λ p (H) = λ(ρ p (H)) for p = 1, 2 is also monotone for H ∈ (0, 1).

λ(r) = 1 2π √ 1 -r 2 R 2 |x 1 + x 2 | |x 1 | + |x 2 | e

Expression and graph of s 2 (H)

From the Delta-method, s Acknowledgements. The authors are grateful to the referee for many relevant suggestions and comments which helped to improve the contents of the paper. The authors also want to thank Antoine Ayache for a discussion, and Mindaugas Stoncelis and Marijus Vaičiulis for providing us with the numerical computations for the graphs in Figures 2, 4 and

Figure 1 .

 1 Figure 1. The graphs of Λ 1 (H) (left) and Λ 2 (H) (right).

Figure 2 .

 2 Figure 2. The graphs of Σp(H), p = 1 (with a pole at 3/4) and p = 2 (with a pole at 7/4) (from Stoncelis and Vaičiulis, 2008, with kind permission of the authors)

  proving the second relation in (2.10) and the proposition.

Property 4 . 1

 41 Assumptions (A.1), (A.1) ′ respectively imply that, for any j ∈ Z and p = 1, 2,

1 0

 1 (i) Let X satisfy assumptions of Theorem 4.2 and α ∈ C p [0, 1], β ∈ C p [0, 1]. Then the statement of Theorem 4.2 holds with X replaced by Z. (ii) Let X satisfy assumptions of Theorem 4.1 and α ∈ C 1 [0, 1], β ∈ C 1 [0, 1]. Then the statement of Theorem 4.1 holds with X replaced by Z. Remark 4.1 A version of the central limit theorem in (4.21) is established in Bardet and Surgailis (2009) with Λ p (H(t))dt replaced by ER p,n under weaker assumption than (A.1) ′ or even (A.1): only properties (4.11)-(4.12) (for p = 1) and (4.13)-(4.14) (for p = 2), in addition to (A.2) p with θ = 0 , are required.

Proposition 5 . 1

 51 Let there exists a limit lim n→∞ E R2,n = Λ.(5.2)Then R2,n a.s.

c

  Z α with c depending on c, and (5.2),(5.3) hold with

Figure 4 .

 4 Figure 4. The graphs of α → Λ(α) = E |Z (1) α +Z (2) α | |Z (1) α |+|Z (2) α |(left) and α → σ(α) (right) for a process with independent increments.

( 1 +

 1 (4.10) also follows by the same argument and the fact that c(t * )c(t) = u n / √ u n / √ n) hold in view of Assumption (A.1)'. Property 4.1 is proved.

(

  i) Denote Zt := α(t)X t , R n,p := R n,p (Z), Rn,p := R n,p ( Z). Clearly, part (i) (the CLT in (4.21) for trended process Z = αX + β) follows from the following relations:√n Rp,n -E Rp,n )

0 ( 1 - 0 ( 1 -

 0101 ) ∈ R 2 be a Gaussian vector with zero mean, unit variances and a correlation coefficient ρ, |ρ| < 1.Then for any b1 , b 2 ∈ R and any 1/2 < a 1 , a 2 < 2 Eψ(a 1 U 1 + b 1 , a 2 U 2 + b 2 ) -Eψ(a 1 U 1 , a 2 U 2 ) ≤ C(b 2 1 + b 2 2 ),(6.23)EU i ψ(a 1 U 1 + b 1 , a 2 U 2 + b 2 ) ≤ C(|b 1 | + |b 2 |), i = 1,2, (6.24) Proof of Corollary 4.2 (a) The argument at the end of the proof of Property 4.1 shows that V satisfies Assumption (A.1) ′ , while (A.2) p follows from Property 4.2. Then the central limit theorem in (4.22) follows from Theorem 4.2. (b) In this case, (A.2) p follows from Property 4.3. Instead of verifying (A.1) ′ , it is simpler to directly verify condition (4.10) which suffices for the validity of the statement of Theorem 4.2. Using f(ξ) = cξ -2H-1 1 + o(ξ -1/2 ) (ξ → ∞),similarly as in the proof of Property 4.3 for j ∈ N * one obtains√ n n -2H E ∆ p,n 0 X ∆ p,n j Xc 2 1+p ∞ cos(x)) p cos(xj) x -2H-1 dx = 2 1+p ∞ cos(x)) p cos(xj) × √ n n 2H+1 (f (nx)c(nx) -2H-1 ) dx -→ n→∞ 0 by the Lebesgue dominated convergence theorem, since ∞ 0

K 1 , 0 Re(e iu v - 1 )dK 1 (

 1011 f n (θ)g α (θ) = g α (θ) e -2n -1 In(θ) -1 ,where, with v := θn 1/α ,I n (θ) := ∞ =: I 1 + I 2 .

|I 1 |K 1 2 1/v 0 u|K 1 1 0/v 1 u 1 -

 11201111 = K 1 (1/v)Re(e i -1) -(u) v sin(uv) du ≤ 2|K 1 (1/v)| + v (u)|du = O(v δ ) + Cv 2 uu -(α-β)+ du + 1δ du = O(v δ∧2 ).

2 2 ) dx 1 dx 2 = √ 1 -r 2 π π 0 |0

 2210 cos φ + sin φ| (| cos φ| + | sin φ|)(1r sin(2φ)) dφ =: I 1 + I 2 , cos φsin φ (cos φ + sin φ)(1 + r sin(2φ)

  where C is nonrandom and depends on the constant K in (3.9) only. Let X t := g(t, B(t)), where B is a standard Brownian motion and g(t, x) is a (jointly) continuous function on [0, 1] × R, having continuous partial derivatives g t (t, x) := ∂g(t, x)/∂t, g x (t, x) := ∂g(t, x)/∂x, g xx (t, x) = ∂ 2 g(t, x)/∂x 2 . By Itô's lemma,

	Example 3.2 dX t = g x (t, B(t))dB(t) + g t (t, B(t)) +	1 2	g xx (t, B(t)) dt,

so that X admits the representation (3.1) with a t = g x (t, B(t)), b t = g t (t, B(t)) + 1 2 g xx (t, B(t)) and the same filtration as in the previous example. Assume that

  1) There exist continuous functions H(t) ∈ (0, 1) and c(t) > 0 for t ∈ [0, 1] such that ∀j ∈ N *

		lim n→∞	sup t∈(0,1)	E X ([nt]+j)/n -X [nt]/n (j/n) 2H(t)	2	-c(t) = 0, with	(4.4)
						lim n→∞	sup t∈(0,1)	H(t) -H(t +	1 n	) log n = 0.	(4.5)
	(A.1) ′ There exist continuous functions H(t) ∈ (0, 1) and c(t) > 0 for t ∈ [0, 1] such that ∀j ∈ N *
		lim n→∞	sup t∈(0,1)	√ n	E X ([nt]+j)/n -X [nt]/n (j/n) 2H(t)	2	-c(t) = 0, with	(4.6)
	lim n→∞	sup t∈(0,1)

  4 because of the requirement γ > 1/2. However, for X = B H , (A.2) 1 holds with appropriate θ > 0 in the wider region 0 < H < 7/8, by choosing θ < 2 -2H arbitrarily close to 2 -2H and then γ < 2 -2H + θ arbitrary close to 4 -4H. A similar choice of parameters θ and γ allows to satisfy (A.2) p for more general X with stationary increments and variogram V (t) ∼ ct 2H (t → 0), under additional regularity conditions on V (t) (see below). Let X have stationary increments and variogram V (t) ∼ ct 2H (t → 0), with c > 0, H ∈ (0, 1). (i) Assume, in addition, that 0 < H < 7/8 and |V ′′ (t)| ≤ Ct -κ (0 < t < 1), for some C > 0 and 4 -4H > κ ≥ 2 -2H, κ > 1/2. Then assumption (A.2) 1 holds.

	Property 4.2

  B H(t) . Therefore Proposition 2.2 ensures the convergence (in L 2 ) of the statistics R p,n to below, using a general moment bound for functions of multivariate Gaussian processes (see Lemma 1 in Section 6). A sketch of this proof can be found in Section 6. Theorem 4.1 Let X be a Gaussian process satisfying Assumptions (A.1) and (A.2) p . Then,

	R p,n a.s. -→ n→∞	0	1	Λ p (H(t))dt	(p = 1, 2).	(4.19)
	Corollary 4.1 Assume that X is a Gaussian process having stationary increments, whose variogram satisfies
	the conditions of Properties 4.2 or 4.3. Then
	R p,n a.s. -→ n→∞	Λ p (H)	(p = 1, 2).	(4.20)

1 0 Λ p (H(t))dt. Bardet and Surgailis (2009) proved a.s. convergence in Theorem 4.1,

  2,n ) is an estimator of the parameter H following a central limit theorem with a convergence rate √ n under semi-parametric assumptions. Similar results were obtained by[START_REF] Guyon | Convergence en loi des H-variations d'un processus gaussien stationnaire[END_REF] and[START_REF] Istas | Quadratic variations and estimation of the local Hölder index of a Gaussian process[END_REF] for generalized quadratic variations under less general assumptions. In the context of Corollary 4.2 and H ∈ (3/4, 1), we expect that R 1,n follows a nongaussian limit distribution with convergence rate n 2-2H . Figure1exhibits that H → Λ 2 (H) is nearly linear and is well-approximated by .1468H +.5174.

	Remark 4.2 Remark 4.3 Consequently,	1 0 Λ 2 (H(t))dt ≈ .1468 H + .5174, where H =	1 0 H(t)dt is the mean value of the function H(•).

  and Surgailis (2010) and compared to the estimator based on generalized quadratic variations discussed in Benassi et al. (1998) and Coeurjolly (2005).

  .15)

	Sketch of the proof of Theorem 4.2. It suffices to show that
			√	n ER n,p -	0	1	Λ p (H(t))dt -→ n→∞	0	(6.16)
	and	√	n R

p,n -ER p,n ) D -→ n→∞ N 0, 1 0 Σ p (H(τ ))dτ . (

6

.17)

  The central limit theorem in(6.19) follows from Theorem 4.2 and Remark 4.1 since Assumption (A.2) p and the convergences (4.11)-(4.14) (with c(t) replaced by α 2 (t)c(t)) can be easily verified for the process Z = αX.Let us turn to the proof of (6.19)-(6.22). For concreteness, let p = 2 in the rest of the proof. Since ∆2,n k Z = α(k/n)∆ 2,n k X + 2α ′ (k/n)n -1 ∆ 1,n k+1 X + O(1/n 2 )X((k + 2)/n), it follows easily from Assumption (A.1) ′ and (4.10) (for X) that Z = α 2 (t)E ∆ 2,n [nt] X ∆ 2,n j+[nt] X + O(n -2H(t)-1 + n -H(t)-2 )implying (4.10) for Z (with c(t) replaced by α 2 (t)c(t)). Whence and using (4.10) and (6.18), relation(6.20) follows similarly as (6.16) above. The proofs of (6.21)-(6.22) uses the following bounds from Bružaitė and Vaičiulis (2008, Lemma 1). Let (U 1 , U 2

		1
			Σ p (H(τ ))dτ ,	(6.19)
		0
	√ n E Rp,n -√ n ER n,p -E Rn,p -→ 1 0 Λ p (H(t))dt -→ n→∞ n→∞ 0,	0,	(6.20) (6.21)
	n var(R n,p -Rn,p ) -→ n→∞	0.		(6.22)
	E ∆ 2,n [nt] Z ∆ 2,n j+[nt]

  2H+1 ) 2 (1ρ 2 (H)) 1ρ 2 2 (H) log 2log(1 + ρ 2 (H)) 2 2H+2 9 log 2 -3 2H 16 log 3 + 6 2H 4 log(3/2) 2 Σ 2 (H),with the approximated graph (using the numerical values of Σ 2 (H) in[START_REF] Stoncelis | Numerical approximation of some infinite gaussian series and integrals[END_REF]):

	2 2 (H) =	∂ ∂x	(Λ 2 ) -1 (Λ 2 (H))	2	Σ 2 (H) and therefore
	s 2 2 (H) =	π (8 -2

  5.

		3.6
		3.4
		3.2
	Standard deviation s2(H)	2.6 2.8 3
		2.4
		2.2
		2
		1.8	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
								H
	Figure 5. The graph of s 2 2 (H)

* This author thanks the Steklov Institute of St Petersburg (Russia) for a nice and fruitful invitation † Research of this author was supported by the Lithuanian State Science and Studies Foundation grant T-70/09.

where the constant C depends only on ρ and does not depend on a 1 , a 2 , b 1 , b 2 . Using (6.23), Assumption (A.1) ′ and the fact that |∆ 2,n k β| ≤ Cn -2 we obtain ER n,2 -E Rn,2 ≤ Cn -1 n-3 k=0 n H(k/n) n -2 2 = O(n -2 ), (6.25) proving (6.21).

To prove (6.22), write

where standardized increments Ỹ (i) n (k), i = 1, 2 are defined as in (6.9)-(6.10) with X replaced by Z, f n,k are defined in (6.11), and µ (1) n (k

.

Note, the ηn (k)'s and η n (k)'s have the Hermite rank ≥ 2 and ≥ 1, respectively, since the Hermite coefficients of order 1 c

From (6.26) and Cauchy-Schwartz inequality, it follows

This proves (6.22) and completes the proof of part (i).

(ii) Similarly as in the proof of part (i), we shall restrict ourselves to the case p = 2 for concreteness. We use the same notation and the decomposition of R 2,n as in part (i):

where

see (6.27). Here, the a.s. convergence to zero of the first term on the r.h.s. of (6.28) follows from Theorem 4.1 since the conditions of this theorem for Z are easily verified. The convergence to zero of the second term on the r.h.s. of (6.28) follows similarly as in (6.25), with the difference that , then E(Q ′ n ) 2 ≤ Cn -2(1-sup t∈[0,1] H(t)) similarly as in (6.29) above. This proves part (ii) and the proposition.