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A central limit theorem for the sample autocorrelations of a L\'evy driven continuous time moving average process

In this article we consider L\'evy driven continuous time moving average processes observed on a lattice, which are stationary time series. We show asymptotic normality of the sample mean, the sample autocovariances and the sample autocorrelations. A comparison with the classical setting of discrete moving average time series shows that in the last case a correction term should be added to the classical Bartlett formula that yields the asymptotic variance. An application to the asymptotic normality of the estimator of the Hurst exponent of fractional L\'evy processes is also deduced from these results.


Introduction
Statistical models are often written in a continuous time setting for theoretical reasons (e.g. diffusions). But if one wants to estimate the parameters of these models, one usually assumes only the observation of a discrete sample. At this point a very general question, the answer of which depends on the model chosen, is to know if the estimation should not have been performed with an underlying discrete model in the beginning. In this article we will consider this for moving average processes and we refer to the classical moving average time series models as a discrete counterpart of this continuous model.
To be more specific, let L = (L t ) t∈R be a two sided one-dimensional Lévy process, i.e. a stochastic process with independent and stationary increments, càdlàg sample paths and which satisfies L 0 = 0. Assume further that L has finite variance and expectation zero, and let f : R → R be in L 2 (R). Let µ ∈ R. Then the process (X t ) t∈R , given by can be defined in the L 2 sense and is called a continuous time moving average process with mean µ and kernel function f , driven by L. See also [6] for more information on such processes, in particular fractional Lévy processes. The process (X t ) t∈R is then strictly stationary. Equation (1.1) is the natural continuous time analogue of discrete time moving average processes where (Z t ) t∈Z is an independent and identically distributed (i.i.d.) noise sequence with finite variance and expectation zero, and (ψ i ) i∈Z is a square summable sequence of real coefficients. The asymptotic behaviour of the sample mean and sample autocorrelation function of X t in (1.2) has been studied for various cases of noise sequences (Z i ) i∈Z , such as regularly varying noise (cf. Davis and Mikosch [7]), martingale difference sequences (cf. Hannan [9]), or i.i.d. sequences with finite fourth moment or finite variance but more restrictive conditions on the decay of the sequence (ψ i ) i∈Z (cf. Section 7 of Brockwell and Davis [3]). Another approach to obtain limit theorems for sample autocovariances is to prove strong mixing properties of the time series under consideration, and provided it has finite (4 + δ)-moment, use the corresponding central limit theorems (such as in Ibragimov and Linnik [10], Theorem 18.5.3). If even stronger strong mixing conditions hold, then existence of a fourth moment may be enough. Observe however that processes with long memory are often not strongly mixing, and in this paper we are aiming also at applications with respect to the fractional Lévy noise, which is not strongly mixing.
In this paper we shall study the asymptotic behaviour as n → ∞ of the sample mean of the process (X t ) t∈R defined in (1.1) when sampled at (∆n) n∈N , where ∆ > 0 is fixed, and of its sample autocovariance and sample autocorrelation function Under appropriate conditions on f and L, in particular assuming L to have finite fourth moment for the sample autocorrelation functions, it will be shown that X n;∆ and ( ρ n;∆ (∆), . . . , ρ n;∆ (h∆)) are asymptotically normal for each h ∈ N as n → ∞. This is similar to the case of discrete time moving average processes of the form (1.2) with i.i.d. noise, but unlike for those, the asymptotic variance of the sample autocorrelations of model (1.1) will turn out to be given by Bartlett's formula plus an extra term which depends explicitly on the fourth moment of L, and in general this extra term does not vanish. This also shows that the "naive" approach of trying to write the sampled process (X n∆ ) n∈Z as a discrete time moving average process as in (1.2) with i.i.d. noise does not work in general, since for such processes the asymptotic variance would be given by Bartlett's formula only. If µ = 0, then further natural estimators of the autocovariance and autocorrelation are given by ρ * n;∆ (∆h) := γ * n;∆ (∆h)/γ * n;∆ (0), h ∈ {0, . . . , n − 1}, (1.7) and the conditions we have to impose to get asymptotic normality of γ * n;∆ and ρ * n;∆ are less restrictive than those for γ n;∆ and ρ n;∆ . We will be particularly interested in the case when f decays like a polynomial, which is e.g. the case for fractional Lévy noises. For a given Lévy process with expectation zero and finite variance, and a parameter d ∈ (0, 1/2), the (moving average) fractional Lévy process (M t;d ) t∈R with Hurst parameter H := d + 1/2 is given by [11]). A process also called fractional Lévy process was introduced before by Benassi et al. [2], where (x) + = max(x, 0) is replaced by an absolute value in (1.8), Although both processes have different distributions, they enjoy similar properties. For instance the sample paths of both versions are Hölder continuous, have the same pointwise Hölder exponent, and they are both locally self-similar (see [2] for the definition of this local property of their distributions). The corresponding fractional Lévy noises based on increments of length ∆ > 0 are given by Hence the fractional Lévy noise is a Lévy driven moving average process with kernel function While the kernel functions f i d,∆ , i ∈ {1, 2}, do not satisfy the assumptions we will impose for the theorems regarding the sample mean X n;∆ and the sample autocorrelation function ρ n;∆ , for d ∈ (0, 1/4) they do satisfy the assumptions we impose for the asymptotic behaviour of ρ * n;∆ , so that an asymptotically normal estimator of the autocorrelation and hence of the Hurst index can be obtained if d ∈ (0, 1/4). For general d ∈ (0, 1/2), one may take the differenced fractional Lévy ;d , t ∈ R, and our theorems give asymptotically normal estimators for the autocorrelation function of these processes. Please note that asymptotically normal estimators of the Hurst exponent for M 2 t;d are already described in [2] but they use fill-in observations of the sample paths X 2 (k/2 n ) for k = 1, . . . , 2 n − 1. If L is a Brownian motion, then X 1 d = CX 2 , where d = means equality in distribution for processes and C is a constant, is the fractional Brownian motion and it is self-similar. Except in this case, fractional Lévy processes are not self-similar and therefore observations on a grid k/2 n do not yield the same information as the time series X i (t), t ∈ Z.
The paper is organised as follows: in the next section we will derive asymptotic normality of the sample mean. Then, in Section 3 we will derive central limit theorems for the sample autocovariance γ n;∆ and the sample autocorrelation ρ n;∆ , as well as for the related estimators γ * n;∆ and ρ * n;∆ of (1.6) and (1.7). As a byproduct of the asymptotic normality, these quantities are consistent estimators of the autocovariance and autocorrelation. Section 4 presents an application of our results to the estimation of the parameters of fractional Lévy noises, where the underlying Hurst parameter is estimated. We also recall there that fractional Lévy noises are mixing in the ergodic-theoretic sense, and we prove that they fail to be strongly mixing.
Throughout the paper, unless indicated otherwise, L will be a Lévy process with mean zero and finite variance σ 2 = EL 2 1 , and X = (X t ) t∈R denotes the process defined in (1.1) with kernel f ∈ L 2 (R), f : R → R. Its autocovariance at lag h ∈ R will be denoted by where the last equation follows from the Itô isometry. Let us set some notations used in the sequel.
If v is vector or A a matrix the transposed is denoted by v ′ , respectively by A ′ . Convergence in distribution is denoted by d →. The function 1 A for a set A is one for x ∈ A, and vanishing elsewhere. The autocorrelation of X at lag h will be denoted by

Asymptotic normality of the sample mean
The sample mean X n;∆ of the moving average process X of (1.1) behaves like the sample mean of a discrete time moving average process with i.i.d. noise, in the sense that it is asymptotically normal with variance σ 2 ∞ k=−∞ γ(k∆), provided the latter is absolutely summable.
and the sample mean of X ∆ , . . . , X n∆ is asymptotically normal as n → ∞, more precisely Remark 2.2. Throughout the paper the assumption that L has zero mean can be dropped very often. For instance, if f ∈ L 1 (R) ∩ L 2 (R), then the assumption of zero mean of L presents no restriction, for in that case L ′ t := L t − tE(L 1 ), t ∈ R, defines another Lévy process with mean zero and the same variance, and it holds Proof. For simplicity in notation, assume that ∆ = 1, and write F = F 1 . Continue F periodically on R by setting The same calculation without the modulus gives (2.2). The proof for asymptotic normality is now much in the same spirit as for discrete time moving average processes, by reducing the problem to m-dependent sequences first and then applying an appropriate variant of Slutsky's theorem. By subtracting the mean we may assume without loss of generality that are independent for each t ∈ Z. From the central limit theorem for strictly stationary (2m − 1)-dependent sequences (cf. Theorem 6.4.2 in Brockwell and Davis [3]) we then obtain that where we used Theorem 7.1.1 in Brockwell and Davis [3] for the second equality. An application of Chebychef's inequality then shows that lim m→∞ lim sup for every ε > 0. Together with (2.5) and (2.6) this implies the claim by a variant of Slutsky's theorem (cf. [3], Proposition 6.3.9).
Remark 2.3. Let us start with an easy remark on a necessary condition on the kernel f to apply the previous theorem. Obviously Remark 2.4. Unlike for the discrete time moving average process of (1.2), where absolute summability of the autocovariance function is guaranteed by absolute summability of the coefficient sequence, for the continuous time series model (1.1) it is not enough to assume that the kernel satisfies f ∈ L 1 (R) ∩ L 2 (R). An example is given by taking ∆ = 1 and For then the function F 1 is given by and since |f (u)| ≤ 1 for all u ∈ R, this implies also f ∈ L 2 (R). Observe further that for non-negative f , condition (2.1) is indeed necessary and sufficient for absolute summability of the autocovariance function.
3 Asymptotic normality of the sample autocovariance As usual, we consider the stationary process We recall that and first we establish an asymptotic result for Cov (γ * n;∆ (p∆), γ * n;∆ (q∆)).

3)
then we have for each p, q ∈ N lim n→∞ nCov (γ * n;∆ (p∆), γ * n; Proof. For simplicity in notation we assume that ∆ = 1. The general case can be proved analogously or reduced to the case ∆ = 1 by a simple time change. We shall first show that for t, p, h, q ∈ Z E(X t X t+p X t+h+q X t+h+p+q ) To show this, assume first that f is of the form where m ∈ N, ǫ > 0 such that 1/ǫ ∈ N, and ψ i ∈ R, i = −m/ǫ, . . . , m/ǫ. Denote Denote further Then (Z i ) i∈Z is i.i.d. and we have At this point we will need to compute the fourth moment of integrals of the Lévy process. Let us state an elementary result that yields a formula for this moment. (3.7) Proof. If ν is the Lévy measure of L and A its Gaussian variance, then by the Lévy Khintchine formula we get Then E( R φ(s)dL s ) 4 is obtained as the fourth derivative of ξ at u = 0. If we recall that (η −3)σ 4 = R x 4 ν(dx), and σ 2 = A+ R x 2 ν(dx), we get (3.7), after elementary but tedious computations.
To continue with the proof of Proposition 3.1, we now apply (3.7) to the special case where f (s) = 1 (0,ǫ] (s) and we get As shown in the proof of Proposition 7.3.1 in [3], we then have E(X t;m,ǫ X t+p;m,ǫ X t+h+p;m,ǫ X t+h+p+q;m,ǫ ) where γ m,ǫ (u) = E(X 0;m,ǫ X u;m,ǫ ), u ∈ R. By (3.8), so that (3.5) follows for f of the form f = f m,ǫ . Now let f ∈ L 2 (R) ∩ L 4 (R) and X t , t ∈ R, defined by (3.1). Then there is a sequence of functions (f m k ,ǫ k ) k∈N of the form (3.6) such that f m k ,ǫ k converges to f both in L 2 (R) and in L 4 (R) as k → ∞. Then for each fixed t ∈ R, we have that X t;m k ,ǫ k → X t in L 2 (P ) (P the underlying probability measure) as k → ∞, where we used the Itô isometry. Further, by Lemma 3.2, and convergence of f m k ,ǫ k both in L 2 (R) and in L 4 (R), we get convergence of X t;m k ,ǫ k to X t in L 4 (P ). This then shows (3.5), by letting f m k ,ǫ k converge to f both in L 2 (R) and L 4 (R) and observing that γ m k ,ǫ k (u) → γ(u) for each u ∈ R. From (3.5) we conclude that, with p, q ∈ N, Cov (γ * n;1 (p), γ * n;1 (q)) = n −1 where T k = γ(k)γ(k − p + q) + γ(k + q)γ(k − p) Then G r is periodic, and by assumption, G r restricted to [0, 1] is square integrable.
Hence we can estimate (3.10) by The same calculation without the modulus and an application of the dominated convergence theorem to (3.9) then shows (3.4).

Remark 3.3. A sufficient condition for (3.3) is that
see e.g. Theorem 7.2.3 in Christensen [5]. Taking ϕ = f then gives the square summability of the autocovariance functions by (1.12).
(b) This follows as in the proof of Proposition 7.3.4 in [3]. One only has to observe that by Theorem 2.1, √ n X n;1 converges in distribution to a normal random variable as n → ∞. In particular, X n;1 must converge to 0 in probability as n → ∞.
(c) The limit theorem follows as in the proof of Theorem 7.2.1 in [3], and for w ij we have the representation giving the claim.
Remark 3.6. It is easy to check that , in accordance with Bartlett's formula, since then (X t∆ ) t∈Z has a discrete time moving average representation with i.i.d. coefficients.
Remark 3.8. Recently, sophisticated and powerful results on the normal approximation of Poisson functionals using Malliavan calculus have been obtained. E.g., Peccati and Taqqu [15, Theorems 2, 3 and 5] prove a central limit theorem for double Poisson integrals and apply this to a specific quadratic functional of a Lévy driven Ornstein-Uhlenbeck process, and Peccati et al. [14,Section 4] obtain bounds for such limit theorems, to name just of few of some recent publications on this subject. It may be possible to apply the results of [14,15] to obtain another proof of Theorem 3.5 under certain conditions such as finite 6th moment, but we have not investigated this issue further. Note that our proof uses only basic knowledge of stochastic integrals and methods from time series analysis.

An application to fractional Lévy noise
We will now apply the previous results to fractional Lévy processes. Recall from (1.8) and (1.9) that these were denoted by respectively, and the corresponding fractional Lévy noises based on increments of length ∆ > 0 by Hence the fractional Lévy noises are Lévy driven moving average processes with kernel functions respectively. Neither f 1 d,∆ nor f 2 d,∆ are in L 1 (R), so Theorem 2.1 cannot be applied because of Remark 2.3. Please note that for the same reason the assumptions for (b) of Theorem 3.5 and for (3.15) are not fulfilled.
For simplicity in notation we assume ∆ = 1, and drop the subindex ∆. Although the fractional noises X i have different distributions for i = 1 and i = 2, they are both stationary with the autocovariance where C i (d) is a normalising multiplicative constant depending on d. Both processes X i are infinitely divisible and of moving average type, hence we know from [4,8] that (X i t ) t∈Z is mixing in the ergodic-theoretic sense. For fixed h ∈ Z, define the function If T denotes the forward shift operator, then and from Birkhoff's ergodic theorem (e.g. Ash and Gardner [1], Theorems 3.3.6 and 3.3.10) we know that for i = 1, 2, and the convergence is almost sure and in L 1 ([1], Theorems 3.3.6 and 3.3.7). Hence, with γ * n = γ * n;1 as defined in (1.6), Since γ(0) = C i (d)σ 2 and γ(1) = C i (d)σ 2 (2 2d − 1), ρ * n (1) = γ * n (1) γ * n (0) is a strongly consistent estimator for 2 2d − 1. Hence, is a strongly consistent estimator for d.
The question of the asymptotic normality of these estimators arises naturally. There are many classical techniques to show the asymptotic normality of an ergodic stationary sequence by assuming some stronger mixing assumption. As far as we know, they do not work in our setting. To illustrate this point, we would like to show that fractional Lévy noises are not strongly mixing. Let us first recall the definition.
Definition 4.1. Let (X n ) n∈Z be a stationary sequence, and let The sequence (X n ) n∈Z is strongly mixing if lim n→∞ α X (n) = 0.
In our case we know the weak mixing property lim n→∞ |P (A∩B)−P (A)P (B)| = 0 for A ∈ σ(X 0 ), B ∈ σ(X n ), because of [4,8]. There are classical central limit theorems for strongly mixing sequences, see e.g. [12] for an overview. The following result, which is stated as Proposition 34 in [12], will be particularly useful for us. Theorem 4.2. Suppose that (X t ) t∈Z is a mean zero, strongly mixing sequence and that there exists some δ > 0 and a constant K > 0 such that where ⌊x⌋ is the integer part of the real number x. Then
Proof. The proof is only written for the fractional noise X 1 denoted by X but it is similar for X 2 . Equation (4.3) holds since L has finite fourth moment and since the kernel function of fractional noise is in L 2 (R) ∩ L 4 (R), and (4.4) follows from the fact that and Var (M m;d ) = Cm 2d+1 ∀ m ∈ N (4.8) for some constant C. To see (4.5) for δ = 2, we use (4.7) and Observe that and that R f 4 m (s) ds ≤ C ′ m 4d+1 for positive constants C, C ′ , which gives the claim.
A consequence of the previous theorem and lemma is the following negative result. Proof. If fractional Lévy noises were strongly mixing, then (4.6) would follow. Please remark that, since fractional noises are increments, S (n) (t) = M i d (⌊nt⌋) − M i d (0), and (Var ( n i=1 X i )) = Cn 2d+1 by (4.8). Owing to the asymptotic self-similarity of fractional Lévy processes (Proposition 3.1 in [2]), we know that where the limit is a fractional Brownian motion with Hurst exponent d + 1/2. A similar asymptotic self-similarity holds for M 1 d . Hence (4.6) is violated and the corollary is proved by contradiction.
Nevertheless one can apply Theorem 3.5 to get the asymptotic normality of the estimator (4.2). Proof. The proof is only written for the fractional noise X 1 denoted by X but it is similar for X 2 . We shall apply (3.14) to get convergence of √ n(ρ * n (1) − ρ(0)) to a Gaussian random variable. First, observe that and ∞ k=−∞ γ(k) 2 < ∞. The latter inequality is classical for the fractional Gaussian noise, when d < 1/4, and holds for fractional Lévy noises since they have the same autocorrelation as fractional Gaussian noise. This also implies (3.11) by Remark 3.4 since f 1 d,1 ≥ 0. Let us check that g 0 := g 0;1 ∈ L 2 (0, 1). Since it follows for all u ∈ (0, 1) that so that even g 0 ∈ L ∞ ([0, 1]). Hence the assumptions of Theorem 3.5 (a) are fulfilled, and the result follows from (3.14).