Correlation structure, quadratic variations and parameter estimation for the solution to the wave equation with fractional noise

Abstract: We compute the covariance function of the solution to the linear stochastic wave equation with fractional noise in time and white noise in space. We apply our findings to analyze the correlation structure of this Gaussian process and to study the asymptotic behavior in distribution of its spatial quadratic variation. As an application, we construct a consistent estimator for the Hurst parameter.


Introduction
The stochastic wave equation driven by space-time white noise or by a Gaussian noise white in time and spatially colored has been widely studied (see e.g. [13], [9], [5], [10] and the references therein). It constitutes a recognized model for the displacement of a vibrating string under a random perturbation.

M. Khalil and C. A. Tudor
The study of SPDEs in general (and of the stochastic wave equation in particular) driven by a fractional noise in time is more recent and it appeared as a consequence of the stochastic calculus with respect to the fractional Brownian motion (fBm in the sequel) and related processes. We refer, among others, to [1], [6], [24].
In this work, we are concerned with the analysis of the solution to the stochastic linear wave equation driven by an additive Gaussian noise which behaves as a fractional Brownian motion with Hurst parameter H ∈ ( 1 2 , 1) with respect to the time variable and as a Wiener process in space. Our purpose is, firstly, to explicitly compute the correlation structure of the solution in space and secondly, to use it in order to obtain the asymptotic behavior in distribution of its spatial quadratic variation with application to the estimation of the Hurst parameter. We have made a similar analysis of the covariance of the solution to the wave equation with time-space white noise in the work [15]. The usual way to compute the covariance, or the mean square of its increment (in time or in space) of the solution is based (see e.g. [10], [6]) on the Fourier transform of the fundamental solution (or the Green kernel) whose expression does not depend on the dimension d ≥ 1. But we have seen in [15] that, for d = 1, a direct calculation based on the formula for Green kernel of the wave equation (and not on its Fourier transform) leads to new insights and brings new information on the correlation structure of the solution. We will employ the same idea in the fractional case and we are able to obtain a closed formula for the spatial covariance of the solution. This formula is useful and gives several unknown facts concerning the correlation structure of the solution, by showing an interesting link between the solution to the wave equation and the fractional Brownian motion. The covariance formula will then be used to obtain sharp estimates for the distribution of the solution, its moments and its path regularity. Moreover, it is a crucial tool which, combinated with the techniques of the Stein-Malliavin calculus, allows to obtain the limit behavior in distribution of the centered quadratic variation in space of the solution to the wave equation. More precisely, if {u(t, x), t ≥ 0, x ∈ R} denotes the solution of the wave equation (t being the time variable and x the space variable), we will show that the (suitably normalized) sequence converges in distribution, as N → ∞ and when H < 3 4 , to a Gaussian random variable. We also derive the rate of convergence via the recent Stein-Malliavin theory (see [20]). The threshold 3 4 also appears in the case of the fractional Brownian motion: it is well-known that the centered and suitably renormalized quadratic variation of the fractional Brownian motion satisfies a CLT for H < 3 4 , H being the self-similarity index of the fBm. In our case, we point out that the value 3 4 does not constitute a threshold for the self-similarity index of the process u (which is self-similar in time of order H + 1 2 and it is not self-similar in space) but for the self-similarity index of the noise in time. This suggests that the behavior of the noise with respect to the time variable highly affects the behavior of the spatial quadratic variation. This makes an interesting link between the theory of regularity of Gaussian stochastic processes and the estimation theory. It follows from the results in Sections 3 and 5 that the analysis of the sharp behavior of the trajectories of the solution to the fractional-white wave equations has a direct impact on many aspect of the process, including the asymptotic behavior of the Hurst index estimators.
It is also well-known that the quadratic variation constitutes a good tool to construct estimators for the self-similarity order of self-similar stochastic processes. We refer to [7], [29] and the references therein. We will assume that, for some fixed t > 1, the process u(t, x) is observed at discrete time u(t, i N ), i = 0, .., N and, via a standard procedure, we construct an estimator based on the spatial observation of the solution at fixed time t. We refer to Section 5.3 for the statistical interpretation of these discrete observations in the case of the vibrating string model. The behavior of the estimator is strongly related to the behavior of the sequence V N , and hence we are able to prove that the estimator is consistent and asymptotically normal.
Our paper is organized as follows. In Section 2 we present general facts concerning the wave equation with fractional noise in time and white noise in space. In Section 3 we compute the spatial covariance of the solution and we deduce several useful facts concerning its correlation structure. Sections 4 and 5 are devoted to the analysis of the asymptotic behavior of the spatial quadratic variations of the solution, with some applications to parameter estimation. The last section contains the proof of the almost sure CLT for the quadratic variation.

Description of the context
In this paragraph we present some general fact concerning the wave equation, the driving Gaussian noise and the definition of the mild solution of this equation. We also present some results on the existence and basic properties of this solution. These results, although not explicitly stated in the literature, can be derived by using the lines of the proofs in Chapter 2 in [28].
We consider below the linear stochastic wave equation driven by an additive infinite-dimensional Gaussian noise W H : } is a real valued centered Gaussian field, over a given complete filtered probability space (Ω, F, (F t ) t≥0 , P), whose covariance function is: where λ is the d-dimensional Lebesgue measure, B d (R d ) is the set of the λbounded Borel subsets of R d and R H is the covariance function of the fBm with Hurst parameter H, and it is given by: Throughout this paper, we consider the regular case, that is the index H is assumed to be in ( 1 2 , 1). Commonly the noise field W H is called "the fractionalwhite noise", it plainly indicates that its behavior is like the fBm in time and like the Wiener process (white) in space. Thus, its spatial increments are independent while the temporal increments present a stochastic dependence and they are positively correlated when H > 1 2 .

The canonical Hilbert space associated to the noise field
Recall that when H > 1 2 , the covariance function (2.3) of fBm is a non-negative function and it can be represented in the following way:  1 2 , is defined as the closure of the linear space generated by ξ with respect to the inner product ., . H W which is expressed by: is also an isometry between H W and Sp(W H ). The scalar product in H W is given by Correlation and quadratic variations for the frac. wave equation 3643

The mild solution to the wave equation
We denote by S(R d ) the Schwartz space of rapidly decreasing C ∞ test-functions on R d and by S (R d ) its dual, the space of tempered distributions. For f ∈ L 1 (R d ), we mean by Ff the Fourier transform of f : where · denotes the Euclidean norm and "." the Euclidean scalar product over R d . An useful formula in our work is the Plancherel-identity: (2.7) Let G 1 be the fundamental solution (called also the Green function) of the homogeneous wave equation ∂ 2 u ∂t 2 − Δu = 0. It is known that G 1 (t, ·) is a distribution in S (R d ) with rapid decrease. An easy way to define it is via its Fourier transform (see e.g. [26], [16]): with σ t denotes the surface measure on the 3-dimensional sphere of radius t. For more details on the kernel G 1 , see e.g. [13]. The solution of (2.1) in its mild formulation is a square-integrable centered field u = {u(t, x); t ∈ [0, T ], x ∈ R d }, that is defined by: One can refer to [8] for further lecture. We will say that the mild solution exists if the integral (2.10) is well-defined, i.e. if the integrand belongs to the space H W . Along this work C, C 1 and C 2 are arbitrary real constants that may change from one line to another and in such computations, they may depend or not on parameters t, s and H. From now on, we set the notation g t,x (s, y) := We state the following result that gives the necessary and sufficient condition for the existence of the solution and the control of its increments in space. Its proof can be obtained by following the proofs in [1], [6] or [28].
, then there exist two constants 0 < C 2 < C 1 such that: Proof: For the first point, we refer to the proof or Theorem 2.8 in [28] which by followed line by line (see also Corollary 2.7 in [28]). The second point follows by the same lines as the proof of Proposition 2 in [6], which presents a complete demonstration when the noise is a fractional Brownian motion with respect to the time and colored with respect to the space. For our case we take β = d, that ensures to get the desired result.

Remark 1.
• This condition d < 2H + 1 is fulfilled, when the spatial dimension d is 1 or 2.
• The bound (2.12) gives the control the spatial increment of the solution.
However, in the sequel, we will restrict to the situation d = 1 and we will have better estimates for square mean and the spatial increment.

Remark 2.
• In virtue of the Kolmogorov-Centsov theorem, for a fixed t ∈ (0, T ], the solution process has a modification (still denoted by the same notation), whose sample paths x → u(t, x) are almost surely Hölder continuous of exponent δ ∈ (0, H − d−1 2 ). Whereas for δ ≥ H − d−1 2 , we note the lack of Hölder continuity. This extends the case of the space-time white noise for d = 1, in which the spatial-wave-solution with space-time white noise is Hölder continuous of order δ ∈ (0, 1 2 ), and also it coincides with the case of heat equation with white-fractional noise.
• One can also show that the process t → u(t, x) is self-similar of order H + 1 − d 2 , see Proposition 2.15 in [28].

The spatial covariance and some consequences
Here we compute the covariance with respect to the space variable of the mild solution (2.10) in dimension d = 1 and we deduce some consequences on the trajectories of the process. As mentioned in the introduction, we will use use the expression of the Green kernel associated to the wave equation (2.9) instead of its Fourier transform. We are able to obtain an explicit formula that brings new information on the correlation structure of the solution u(t, x) (2.10). This formula is also crucial in order to derive the results in the next section (CLT for the quadratic variation and estimation of the Hurst parameter).

The spatial covariance
From now on, assume that d = 1. We have the following result, which is a key results for our work.

Lemma 1.
Let T > 0 and fix t ∈ (0, T ]. Then for every x, y ∈ R and every H > 1 2 , we have: . Proof: For a fixed t ∈ (0, T ], it can be shown that for every x, y ∈ R, we get: We assume without loss of generality that x ≤ y, and then we start computing the first term R 1 from above. Here two cases can be discussed: Separately we treat the quantities R 1,1 and R 1,2 . For the first one, we make use of the change of variablesṽ = u − v in the integral dv, and we infer that: Note that on the set {u < y−x}, it yields that: Consequently this means that: Looking at the quantity R 1,1,A , it can be shown that: For the second term R 1,1,B , we may calculate the integral and it produces that: Combining (3.5) and (3.6) in (3.4), we infer that: Focusing now on R 1,2 , a modicum of calculus leads to: Plugging the last two quoted results (3.7) and (3.8) in (3.3), we easily obtain that for all x, y ∈ R: with c H := 4H−1 4(2H+1) . Dealing now with the remaining part R 2 of the statement (3.2), and using the change of variables (ũ,ṽ) = (t − u, t − v), we can show that: Proceeding by the same calculus as R 1 , we find that: Therefore, due to the expression (3.9) and (3.10), the final expression of the covariance function R can be written in the following way: and the desired conclusion follows.

Remark 3.
For the classical case when H = 1 2 , we note that for every x, y ∈ R: from one hand R 1,2 = 0 and from another hand: which implies that: and obviously we deduce that it coincides with the covariance's expression for the space-time white case (see [15]).

Remark 4.
Looking at the proceeding expression (3.1), we deduce that the spatial solution (u(t, x)) x∈R is a stationary process, this remains that its behavior is different from the spatial Heat-solution with white-fractional noise, which is self-similar with index H − d−1 2 . Some results due to the expression of the spatial covariance function can be listed below in the following subsection.

Continuity of the covariance
From (3.1), we can infer that: where the function f is defined for all z ∈ R in the next way: Clearly f is a continuous function on R, in particular at the points z = t and z = 2t.

Explicit distribution of the spatial solution
Another interesting consequence is the second moment of the spatial solution, which is for a fixed t > 0 and for every x ∈ R equal to: . (3.12) Recall that the solution process (u(t, x)) x∈R is a centered Gaussian one, hence for fixed t > 0, we may make out that: In particular, the p-moment is: (3.14)

Sharp estimates of the spatial increment
Looking again at the mean square of the spatial increment of the solution, we can find sharp estimates according to the covariance function's expression. In fact we retrieve the result established in Proposition 1, point 2, which is based on using the Fourier transform of the Green kernel and on mimicking the proof evoked in [6], it shows that for d = 1: Yet, according to the covariance (3.1) we can get this result and even more precise bounds with explicit expressions for the constants appearing in the result. Indeed, fix t > 0 and for every x, y ∈ R, we have: For the first situation when |y − x| < t, we get: Which obviously leads to: For the second situation when t ≤ |y − x| < 2t, it implies that: . (3.19) Since in this case we have 1 ≤ |y−x| t < 2, so it yields that: Consequently we deduce that: Actually, in this second case and by the same reasoning we can obtain more regularity for the mean square spatial increment, for instance:

Spatial modulus of continuity
From the above estimates on the spatial increment of the solution, we can deduce useful information on the modulus of continuity of the process (u(t, x)) x∈R given by (2.10) with respect to its space variable. Let us fist recall some general definitions. Let f be an increasing function in R + such that lim x→0 + f (x) = 0. Let (Y t ) t∈I be a stochastic process with index set I ⊂ R and let ρ be a metric on I. We say that the function f is an almost sure uniform modulus of continuity on (I, ρ), if there exists an almost-surely positive random variable α 0 such that for α < α 0 one has sup s,t∈T ;ρ(s,t)<α For (sub)Gaussian processes, there is a wide theory on the modulus of continuity in terms of the covariance structure of the process and the magnitude of its increment (see, among others, [11], [12], [18], [27], [30]). In particular, if Y is a Gaussian process such that for all s, t ∈ I, where G : R + → R + is an increasing function with G(0) = 0, then is an almost sure uniform modulus of continuity for Y on I with respect to the Euclidean metric. From the estimates (3.19) and (3.20) in Section 3.2.3, we can immediately get the modulus of continuity of the Gaussian process (u(t, x)) x∈R . Indeed, fix t > 0. By the bound (3.19) and (3.20), we easily deduce that for every

Relation with the fractional Brownian motion
Another interesting property resulting from the formula (3.1) is that the spatial increments of the solution are related to the increments of the fractional Brownian motion. Indeed, assume t is fixed and x, y are such that |y − x| < t (this will be the case in the next section). In this case, it follows from (3.1) that  1] .

M. Khalil and C. A. Tudor
Since the behavior of our sequence (V N ) N ≥1 (defined below in(4.1)) relies on how big is the correlation of its spatial increments, we will estimate its L 2 (Ω)norm by using essentially the earlier expression (3.1) and some elements of Malliavin calculus.

The spatial quadratic variation
Fix d = 1 and the time t ∈ (0, T ], T > 0. Take an equidistant spatial partition of the unit rectangle [0, 1] such that for every N ≥ 1 and for every j = 0, . . . , N, we designate by x j = j N . The centered renormalized quadratic variation statistic over the unit interval [0, 1], can be defined in the following way: Our purpose is to find the asymptotic behavior of the renormalized partial sum V N as N goes to infinity.
To prove this, a convenient device that we will use the recent Stein-Malliavin theory, see [20]. To this end, we need to introduce the basic tools of the Malliavin calculus.

Malliavin calculus
We assume that the reader has a basic knowledge of such notions from stochastic analysis. Here, we shall only recall some elementary facts; our main reference on this realm is [22]. Consider H a real separable infinite-dimensional Hilbert space with its associated inner product ., . H , and (B(ϕ), ϕ ∈ H) an isonormal Gaussian process on a probability space (Ω, F, P), which is a centered Gaussian family of random variables such that E (B(ϕ)B(ψ)) = ϕ, ψ H , for every ϕ, ψ ∈ H. Denote by I q the qth multiple stochastic integral with respect to B. This I q is actually an isometry between the Hilbert space H q (symmetric tensor product) equipped with the scaled norm 1 √ q! · H ⊗q and the Wiener chaos of order q, which is defined as the closed linear span of the random variables H q (B(ϕ)) where ϕ ∈ H, ϕ H = 1 and H q is the Hermite polynomial of degree q ≥ 1 defined by: The isometry of multiple integrals can be written as: for p, q ≥ 1, f ∈ H ⊗p and g ∈ H ⊗q , It also holds that: wheref denotes the canonical symmetrization of f and it is defined by: in which the sum runs over all permutations σ of {1, . . . , q}.
We recall that any square-integrable random variable F , which is measurable with respect to the σ-algebra generated by B, can be expanded into an orthogonal sum of multiple stochastic integrals: where the series converges in L 2 (Ω)-sense and the kernels f q , belonging to H q , are uniquely determined by F .
We denote by D the Malliavin derivative operator that acts on cylindrical random variables of the form F = g(B(ϕ 1 ), . . . , B(ϕ n )), where n ≥ 1, g : R n → R is a smooth function with compact support and ϕ i ∈ H. This derivative is an element of L 2 (Ω, H) and it is defined as: The operator D is continuous from D α,p (H) into D α−1,p (H) .

Renormalization of V N
To renormalize V N we need to understand the behavior of E V 2 N as N → ∞. To this end, we need a careful analysis of the auto-correlation E (u(t, Applying the expression (3.1), we clearly obtain for every 0 ≤ i, j ≤ N : According to the value of the temporal index t, the former covariance function (4.7) can be written in the following way: • If t ∈ (1, T ], it yields that |i − j + 1| < tN and we get: where ϕ H is given by (3.25) and Recall that ϕ H (k) ∼ k→∞ 2H(2H − 1)|k| 2H−2 , with the symbol "∼" means that both sides have the same limit as k goes to ∞.
As mentioned in the subsection 3.2.5, this means that, when the time is large enough, the increment of the spatial wave solution behaves as the sum of the increments generated by the fBm B H modulo a constant and the fBm B H+ 1 2 modulo a constant, with B H+ 1 2 independent by B H . • If t ∈ (0, 1], several situations can be evoked according to the values of tN and also of 2tN , that is why it seems to be quiet hard to estimate the L 2 (Ω)norm of the increments by using our approach basing on the expression of the covariance function (4.7).
For the rest of the work we adopt that the time t ∈ (1, T ]. Due to the Subsection 4.1, we denote by H the canonical Hilbert space associated to the Gaussian solution process (u(t, x)) x∈ [0,1] . This Hilbert space is defined as the closure of the set ξ of indicator functions 1 [0,x] , x > 0, with respect to the inner product: x)u(t, y) , for a fixed t ∈ (1, T ]. The Gaussian space generated by (u(t, x)) x∈ [0,1] , t ∈ (1, T ], can be identified with an isonormal Gaussian process of the type (X(h)) h∈H . We also designate by I q , q ≥ 1 the multiple Wiener-integral with respect to the Gaussian process (u(t, x)) x∈ [0,1] , so the increment u(t, y) − u(t, x) can be expressed as I 1 (1 [x,y] ), for every x < y.
Define now the next sequence: Using again (4.7) when t ∈ (1, T ], it implies that: (4.11) and consequently, for N ≥ 1, F N can be re-written as: The next lemma shows that the deterministic sequence v N > 0 converges, as N goes to ∞, to a strictly positive constant. It will play no role in the limit behavior of (F N ) N ≥1 , it is just used to normalize and to guarantee that E(F N 2 ) = 1.

M. Khalil and C. A. Tudor
(4.13) It turns out to check that the first term T 1 is the dominant one, and it converges to a finite limit, while the other terms are negligible. So, by using the dominated convergence theorem, we get that: (4.14) Recall that the function ϕ H (k) behaves, for k large enough, as H(2H−2)|k| 2H−2 , which yields that the series k∈Z (ϕ H (k)) 2 is finite only if H < 3 4 . Also, as ϕ H (0) = 1, it is obvious that C 0 ∈ (0, ∞).
Moving to the second term T 2 , we obtain: which is true due to the Cesàro-Lemma and the fact that H < 3 4 .
Similarly as for the second term, we deduce that: By putting together (4.14), (4.15) and (4.16), the proof is completed.

Remark 6. The appearance of the threshold H = 3 4 is interesting. Note that H represents the self-similarity index of the noise (in time) and not of the solution.
This is related to the observation noticed in the Section 3.2.5: the main part of V N comes from the fBm B H . A similar phenomenon has been noticed in [25].

Central Limit Theorem and rate of convergence
In order to verify the CLT for the sequence (V N ) N ≥1 , our main tool will be the Theorem 5.2.6 in [20], which provides a description of the normal approximation of multiple stochastic integrals, by the aid of explicit bounds of the well-known distances d (Kolmogorov, Total Variation, Wasserstein). This result is based on the classical Berry-Esseen inequality.
be a sequence of random variables belonging to the qth Wiener chaos such that: Then, F N converges in law to Z ∼ N (0, 1) if and only if Furthermore,

Main result
Our first main result is summarized in the following lemma.

Proof:
We apply Theorem 1. Since for all N ≥ 1: it is a enough to estimate the quantity: From Subsection 4.1, we get: which obviously gives: We designate by P 1 the product of four elements that are the functions ϕ H . As well, P 2 contains all the terms with the form of a product of only one function ϕ H and three functions ϕ H+ 1 2 . P 3 is composed of all the terms which are the product of two functions ϕ H and two functions ϕ H+ 1 2 . P 4 contains all the terms which are written as a product of three functions ϕ H and one function ϕ H+ 1 2 , and finally P 5 is the product of four functions ϕ H+ 1 2 . The biggest term in (5.2) is P 1 , straightforward calculations show that all the other terms are negligible. Typically referred to the proof of the Theorem 7.3.1 in [20], we set, for all N ≥ 1, the functions: Also, for two sequences (u(n), n ∈ Z) and (v(n), n ∈ Z), we define their convolution by We will need the Young's inequality if s, p, q ≥ 1 with 1 s + 1 = 1 p + 1 q . We can write: By the aid of Young's inequality (5.4) for s = 2 and p = q = 4 3 , we obtain: As the function ϕ H behaves like C|k| 2H−2 when |k| goes to ∞, it yields that: Hence, recall that v n ∈ (0, ∞) only for H ∈ ( 1 2 , 3 4 ), we deduce the following result: 3 4 ). (5.6) The same reasoning will be used for the other terms, giving: Due to the bounds in (5.5), and the asymptotic behavior of ϕ H+ 1 2 which shows that 1 19 4 , we get for N ≥ 3: 3 4 ). (5.7) Similarly, and since 1 , we can write that for N ≥ 3: 3 4 ).
which checked the desired bounds.

Remark 7.
• The previous result has a discrete version which is mentioned in the Theorem 7.3.1 of [20], with the stationary Gaussian sequence , this sequence indicates the normalized noise generated by the spatial wave solution such that v t (k) := u(t, k + 1) − u(t, k), k ∈ Z.
• A concrete interpretation is there exist a remarkable impact of the fractional Brownian motion on the spatial increments of the wave-solution, the result that we obtained shows that when, H ∈ ( 1 2 , 3 4 ), the quadratic variation of B H and the spatial quadratic variation of the wave solution have the same rate of convergence to the normal distribution, although the fractional part of the noise is temporal and all our work is done in the spatial case. This reinforces the idea that the study of Gaussian processes does not necessarily rely on the analysis of filtrations, Markovian aspects etc. 5 8 is the same as in case of the fBm (see [20]). The same result as (5.1), modulo a change of the constant, holds for the different distances between the laws of random variables (e.g. Kolmogorov, Total Variations, Wasserstein . . . ).

Optimal rate of convergence
In the case of the total variation distance (denoted d T V in the sequel), we can obtain better estimates for the distance between F N and the standard normal law than those in (5.1). This can be done via the main result in [21] (see also [19]) where is proved that the optimal rate of convergence to the standard normal law of a sequence (F N ) N ≥1 in the qth Wiener chaos, under the distance d T V , is given by the quantity where u n ∝ v n means that 0 < lim n→∞ u n v n < ∞. Based on this result, we obtain the following sharp estimate for the rate of convergence of (4.10).

(5.12)
Proof: Recall that We need to estimate the third moment of the sequence F N . By using the product formula (4.5) and the isometry property (4.3), we have Using the definition of the contraction (4.6), As in the proof of Lemma 3, the dominant part of with ϕ H given by (3.25). So This above quantity in the right-hand side already appeared in the case of the quadratic variations of the fBm. From [3] especially relation (6.57) (or Proposition 4.2 in [21]), we deduce the following 3 4 ).

Estimation of the Hurst parameter H
The variations of a stochastic process play a crucial role in its probabilistic and statistical analysis. Amomg others, its asymptotic behavior is a fundamental tool in estimation theory.
We will show in this subsection that the asymptotic behavior of the sequence (V N ) N ≥1 defined in (4.1), is related to the asymptotic properties of a class of estimators for the Hurst parameter H. The construction has been used in [7] for the case of the fBm and in [28] and the references therein, for other selfsimilar processes.
The idea for estimating the parameter H of the fractional noise driven our wave equation, is based on the discrete spatial observations of the solution. As standard method is to construct an estimator based on the spatial quadratic variations of the process (u(t, x)) x∈ [0,1] . For fixed t ∈ (1, T ], we observe u(t, i N ), i = 0, . . . , N, and we define the next sequence: This implies that: with k 1 , k 2 from (4.9) and thus Estimating A N by S N , we can construct the following estimator: An important relation that we can easily establish between V N and S N , for all N ≥ 1, is at this form: So using the fact that log(1 + x) ∼ x, for x close to 0, it gives that (because V N √ N converges almost surely to 0 as N → ∞, see [28], Section 5.5): Moreover, H N is an asymptotically normal estimator such that: Proof: Viewing the relation (5.21) and recalling that due to the Borel-Cantelli lemma, the sequence (V N ) N ≥ converges almost surely to 0, as N goes to ∞, we obtain the strong consistency of the estimator H N .
The second part of the theorem is deduced from Lemma 3 in which we already verified the asymptotic normality of the sequence (V N ) N ≥1 .

Statistical interpretation
Consider a tightly stretched string without slope and let x a point on the string at t = 0, i.e. in the equilibrium position. When the string vibrates, we can assume that the horizontal displacement of the point x is negligible, since there is no slope. So u(t, x) represents the position at time t of the point x on the vibrating string, under a random force W H (the fractional-white Gaussian noise).
Consequently, observing u(t, x i ), i = 0, .., N means that at a certain moment t > 1, we are able to observe the position of the string at x i , which seems reasonable from the practical point of view. It is also worth to note that only from the discrete observation at one arbitrary time, it is possible to get the estimation of the Hurst parameter H and implicitly the estimation of the self-similarity index of the process (u(t, x), t ≥ 0) which is H + 1 2 . Moreover, as showed before, the estimator is strongly consistent and asymptotically normal.
As a final comment, let us notice that the parameter H characterizes the main properties of the process u is time and in space, see e.g. Lemma 2.12 and Remark 2.
In the next section we present an improvement of the CLT which is the ASCLT.

Almost Sure Central Limit Theorem
The ASCLT was stated firstly by Lévy [17], then it was extensively treated by many other authors, for instance in [14]. It constitutes an improvement of the CLT. In the case of multiple stochastic integrals of fixed order q ≥ 2 we have the following result from [2] (recall that the contraction is defined in (4.6)): Theorem 3. Fix q ≥ 2, and let (F N ) N ≥1 be a sequence of random variables defined by F N := (I q (f N )) N ≥1 with f N ∈ H q , such that for all N ≥ 1, E(F 2 N ) = q! f N 2 H ⊗q = 1 and f N ⊗ r f N H ⊗2(q−r) goes to 0, as N goes to ∞, for every r = 1, . . . , q − 1. Then, F N L − → Z ∼ N(0, 1), as N → ∞. Moreover, if the following two conditions are fulfilled: then (F N ) N ≥1 satisfies an ASCLT. In other words, almost surely, for all bounded and continuous function ϕ : R → R, The main result of this section is summarized in the following proposition.
Proof: Using Theorem 3, we only need to check the hypothesis (A 1 ) and (A 2 ), so from one hand we have: which leads to: