SELF-SIMILARITY PARAMETER ESTIMATION AND REPRODUCTION PROPERTY FOR NON-GAUSSIAN HERMITE PROCESSES

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1 Communications on Stochastic Analysis Vol. 5, o Serials Publications SELF-SIMILARITY PARAMETER ESTIMATIO AD REPRODUCTIO PROPERTY FOR O-GAUSSIA HERMITE PROCESSES ALEXADRA CHROOPOULOU*, CIPRIA A. TUDOR, AD FREDERI G. VIES Abstract. Let Z q,h t t,1 be a Hermite processes of order q and with Hurst parameter H 1, 1. This process is H-self-similar, it has stationary increments and it exhibits long-range dependence. This class contains the fractional Brownian motion for q = 1 and the Rosenblatt process for q =. We study in this paper the variations of Z q,h by using multiple Wiener -Itô stochastic integrals and Malliavin calculus. We prove a reproduction property for this class of processes in the sense that the terms appearing in the chaotic decomposition of the their variations give birth to other Hermite processes of different orders and with different Hurst parameters. We apply our results to construct a consistent estimator for the self-similarity parameter from discrete observations of a Hermite process. 1. Introduction 1.1. Background and motivation. The variations of a stochastic process play a crucial role in its probabilistic and statistical analysis. Best known is the quadratic variation of a semimartingale, whose role is crucial in Itô s formula for semimartingales; this example also has a direct utility in practice, in estimating unknown parameters, such as volatility in financial models, in the so-called historical context. For self-similar stochastic processes the study of their variations constitutes a fundamental tool to construct good estimators for the self-similarity parameter. These processes are well suited to model various phenomena where long memory is an important factor internet traffic, hydrology, econometrics, among others. The most important modeling task is then to determine or estimate the self-similarity parameter, because it is also typically responsible for the process s long memory and other regularity properties. Consequently, estimating this parameter represents an important research direction in theory and practice. Several approaches, such as wavelets, variations, maximum likelihood methods, have been proposed. We refer to the monograph 1 for a complete exposition. The family of Gaussian processes known as fractional Brownian motion fbm is Received Mathematics Subject Classification. Primary 6F5; Secondary 6H5, 6G18. Key words and phrases. Multiple stochastic integrals, Malliavin calculus, Hermite process, fractional Brownian motion, non-central limit theorems, quadratic variations, bootstrap. * Chronopoulou is partially supported by SF grant DMS Tudor is Associate member of the team SAMM, Université de Panthéon-Sorbonne Paris 1. Viens is partially supported by SF grants DMS and

2 16 ALEXADRA CHROOPOULOU, CIPRIA A. TUDOR, AD FREDERI G. VIES particularly interesting, and most popular among self-similar processes, because of fbm s stationary increments, its clear similarities and differences with standard Brownian motion, and the fact that its self-similarity parameter H, known as the Hurst parameter, is also clearly interpreted as the memory length parameter the correlation of unit length increments n time units apart decays slowly at the speed n H, and the regularity parameter fbm is α-hölder continuous on any bounded time interval for any α < H. Soon after fbm s inception, the study of its variations began in the 197 s and 198 s; of interest to us in the present article are several such studies of variations which uncovered a generalization of fbm to non- Gaussian processes known as the Rosenblatt process and other Hermite processes:, 5, 7, 17 or 18. We briefly recall some relevant basic facts. We consider Bt H t,1 a fractional Brownian motion with Hurst parameter H, 1. As such, B H is the continuous centered Gaussian process with covariance function R H t, s = E Bt H Bs H 1 = s H + t H t s H, s, t, 1. B Equivalently, B H H = and E s Bt H = t s H. It is the only Gaussian process H which is self-similar with stationary increments. Consider = t < t 1 <... < t = 1 a partition of the interval, 1 with t i = i for i =,..., and define the following q variations v q 1 = i= H q Bt H B H 1 t i 1 E Bt H Bt H = i= i H q B H t B H ti t t i H where H q with q represents the Hermite polynomial of degree q. Then it follows from the above references that for: for H < 1 1 q the limit in distribution of 1 v q is a centered Gaussian random variable, for H = 1 1 q the limit in distribution of log 1 v q is a centered Gaussian random variable, for 1 1 q < H < 1 the limit of q1 H 1 v q is a Hermite random variable of order q with self-similarity parameter qh This latter random variable is non-gaussian; it equals the value at time 1 of a Hermite process, which is a stochastic process in the qth Wiener chaos with the same covariance structure as fbm; as such, it has stationary increments and shares the same self-similarity, regularity, and long memory properties as fbm; see Definition.1. We also mention that very recently, various interesting results have been proven for weighted power variations of stochastic processes such as fractional Brownian motion see 11, fractional Brownian sheets see 15, iterated Brownian motion see 13 or the solution of the stochastic heat equation see 16 or 3. Because of a natural coupling, the last limit above also holds in L Ω see 1. In the critical case H = 1 1 q the limit is still Gaussian but the normalization involves a logarithm. These results are widely applied to estimation problems; to avoid the barrier H = 3 4 that occurs in the case q =, one can use

3 ESTIMATIO AD REPRODUCTIO FOR HERMITE PROCESSES 163 higher-order filters, which means that the increments of the fbm are replaced by higher-order increments, such as discrete versions of higher-order derivatives, in order to obtain a Gaussian limit for any H see 9, 8, 4. The appearance of Hermite random variables in the above non-central limit theorems begs the study of Hermite processes as such. Their practical aspects are striking: they provide a wide class of processes from which to model long memory, self-similarity, and Hölder-regularity, allowing significant deviation from fbm and other Gaussian processes, without having to invoke non-linear stochastic differential equations based on fbm, and the notorious issues associated with them see 14. Just as in the case of fbm, the estimation of the Hermite process s parameter H is crucial for proper modeling; to our knowledge it has not been treated in the literature. We choose to tackle this issue by using variations methods, to find out how the above central and non-central limit theorems fit in a larger picture. 1.. Main results: summary and discussion. In this article, we show results that are interesting from a theoretical viewpoint, such as the reproduction properties the variations of Hermite processes give birth to other Hermite processes; and we provide an application to parameter estimation, in which care is taken to show that the estimators can be evaluated practically. Let Z q,h be a Hermite process of order q with selfsimilarity parameter H 1, 1 as defined in Definition.1. Define the centered quadratic variation statistic V := 1 1 i= Z q,h E Z q,h Z q,h i Z q,h i Also for H 1/, 1, and q \ {}, we define a constant which will recur throughout this article: HH 1 1/ dh, q : = q!h H 1 q, 1/ H = 1 + H q We prove that, under suitable normalization, this sequence converges in L Ω to a Rosenblatt random variable. Theorem 1.1. Let H 1/, 1 and q \{}. Let Z q,h be a Hermite process of order q and self-similarity parameter H see Definition.1. Let c be an explicit positive constant it is defined in Proposition 3.1. Then c H/q V converges in L Ω to a standard Rosenblatt random variable with parameter H := H 1 q + 1, that is, to the value at time 1 of a Hermite process of order and self-similarity parameter H. The Rosenblatt random variable is a double integral with respect to the same Wiener process used to define the Hermite process; it is thus an element of the second Wiener chaos. Our result shows that fbm is the only Hermite process for which there exists a range of parameters allowing normal convergence of the quadratic variation, while for all other Hermite processes, convergence to a second chaos random variable is universal. Our proofs are based on chaos expansions

4 164 ALEXADRA CHROOPOULOU, CIPRIA A. TUDOR, AD FREDERI G. VIES into multiple Wiener integrals and Malliavin calculus. The main line of the proof is as follows: since the variable Z q,h t is an element of the qth Wiener chaos, the product formula for multiple integrals implies that the statistics V can be decomposed into a sum of multiple integrals from the order to the order q. The dominant term in this decomposition, which gives the final renormalization order H/q, is the term which is a double Wiener integral, and one proves it always converges to a Rosenblatt random variable; all other terms are of much lower orders, which is why the only remaining term, after renormalization, converges to a second chaos random variable. The difference with the fbm case comes from the limit of the term of order, which in that case is sometimes Gaussian and sometimes Rosenblatt-distributed, depending on the value of H. We also study the limits of the other terms in the decomposition of V, those of order higher than, and we obtain interesting facts: all these terms, except the term of highest order q, have limits which are Hermite random variables of various orders and selfsimilarity parameters. We call this the reproduction property for Hermite processes, because from one Hermite process of order q, one can reconstruct other Hermite processes of all lower orders. The exception to this rule is that the normalized term of highest order q converges to a Hermite r.v. of order q if H > 3/4, but converges to Gaussian limit if H 1/, 3/4. Summarizing, we have the following. Theorem 1.. Let Z q,h be again a Hermite process, as in the previous theorem. Let T n be the term of order n in the Wiener chaos expansion of V : this is a multiple Wiener integral of order n, and we write V = q n=1 c nt n where c q k = k! q. k For every H 1/, 1 and for every k = 1,..., q 1 we have convergence of the expression z k,h 1 H k T k in L Ω to Z r,k, a Hermite random variable of order r = k with self-similarity parameter K = kh 1 + 1, where z k,h is a constant. For every H 1/, 3/4, with k = q, we have convergence of x 1/ 1,H,q Tq to a standard normal distribution, with x 1,H,q a positive constant depending on H and q. For every H 3/4, 1, with k = q, we have convergence of x 1/,H,q H T q in L Ω to Z q,h 1, a Hermite r.v. of order q with parameter H 1; with x,h,q a positive constant depending on H and q. For H = 3/4, with k = q, we have convergence of / log x 1/ 3,H,q T q to a standard normal distribution, with x 3,H a positive constant depending on H and q. Some of the aspects of this theorem had been discovered in the case of q = Rosenblatt process in. In that paper, the existence of a higher-chaos-order term with normal convergence had been discovered for the Rosenblatt process with H < 3/4, while the case of H 3/4 had not been studied. The entire spectrum of convergences in Theorem 1. was not apparent in, however, because it was unclear whether the term T s convergence to a Rosenblatt r.v. was due to the fact that we were dealing with input coming from a Rosenblatt process, or whether it was a more general function of the structure of the second Wiener

5 ESTIMATIO AD REPRODUCTIO FOR HERMITE PROCESSES 165 chaos; here we see that the second alternative is true. The paper also exhibited a remarkable structure of the Rosenblatt data when H < 3/4. In that case, as we see in Theorem 1., there are only two terms in the expansion of V, T and T 4 ; moreover, and this is the remarkable feature, the proper normalization of the term T converges to none other than the Rosenblatt r.v. at time 1. Since this value is part of the observed data, one can subtract it to take advantage of the Gaussian limit of the renormalized T 4, including an application to parameter estimation in. In Theorem 1. above, if q >, even if H 3/4, by which a Gaussian limit can be constructed from the renormalized T q, we still have at least two other terms T, T 4, T q, and all but at most one of these will converge in L Ω to Hermite processes with different orders, all different from q, which implies that they are not directly observed. This means our Theorem 1. proves that the operation performed with Rosenblatt data, subtracting an observed quantity to isolate T q and its Gaussian asymptotics, is not possible with any higher-order Hermite processes. The last aspect of this paper applies Theorem 1.1 to estimating the parameter H. Let S be the empirical mean of the individual squared increments S = 1 1 i= Z q,h Z q,h i and let Ĥ = log S / log. We show that Ĥ is a strongly consistent estimator of H, and we show asymptotic Rosenblatt distribution for H 1 H/q Ĥ log. The fact that this estimator fails to be asymptotically normal is not a problem in itself. What is more problematic is the fact that if one tries to check ones assumptions on the data by comparing the asymptotics of Ĥ with a Rosenblatt distribution, one has to know something about the normalization constant 1 H/q. Here, we prove in addition that one may replace H in this formula by Ĥ, so that the asymptotic properties of Ĥ can actually be checked. Theorem 1.3. The estimator Ĥ is strongly consistent, i.e. lim Ĥ = H almost surely. Moreover there exists a standard Rosenblatt random variable R with self-similarity parameter 1 + H 1/q such that lim E 1 Ĥ/q H Ĥ Here a H = 1 + H 1/q 1 + H 1/q. log c c 1/ 1,H R =, Replacing the constant c 1,H by its value in terms of Ĥ instead of H seems to lead to the above convergence. However, this article does not present any numerical results illustrating model validation based on the above theorem; moreover such applications would be much more sensitive to the convergence speed than to the actual constants; therefore we omit the proof of this further improvement on c 1,H.. Preliminaries.1. Multiplication in Wiener Chaos. Let W t t,1 be a classical Wiener process on a standard Wiener space Ω, F, P. If f L, 1 m with m 1

6 166 ALEXADRA CHROOPOULOU, CIPRIA A. TUDOR, AD FREDERI G. VIES integer, we introduce the multiple Wiener-Itô integral of f with respect to W. We refer to 14 for a detailed exposition of the construction and the properties of multiple Wiener-Itô integrals. Let f S, i.e. f is an elementary function, meaning that f = c i1,...i m 1 Aii... A i m i 1,...,i m where i 1,..., i m describes a finite set and the coefficients satisfy c i1,...i m = if two indices i k and i l are equal and the sets A i B, 1 are disjoints. For such a step function f we define I m f = c i1,...i m W A i1... W A im i 1,...,i m where we put W a, b = W b W a. It can be seen that the application I m constructed above from S to L Ω satisfies E I n fi m g = n! f, g L,1 n if m = n.1 and if m n. It also holds that I n f = I n f where f denotes the symmetrization of f defined by fx 1,..., x n = 1 n! σ n fx σ1,..., x σn. Since the set S is dense in L, 1 n the mapping I n can be extended to a linear continuous operator from from L, 1 n to L Ω and the above properties hold true for this extension. ote also that I n f with f symmetric can be viewed as an iterated stochastic integral I n f = n! 1 tn t... ft 1,..., t n dw t1... dw tn ; here the integrals are of Itô type; this formula is easy to show for elementary f s, and follows for general symmetric function f L, 1 n by a density argument. We recall the product for two multiple integrals see 14: if f L, 1 n and g L, 1 m are symmetric then it holds that I n fi m g = m n l= m l! l n l I m+n l f l g. where the contraction f l g belongs to L, 1 m+n l for l =, 1,..., m n and it is given by f l gs 1,..., s n l, t 1,..., t m l =,1 l fs 1,..., s n l, u 1,..., u l gt 1,..., t m l, u 1,..., u l du 1... du l..3.. The Hermite process. Recall that the fractional Brownian motion process Bt H t,1 with Hurst parameter H, 1 can be written as B H t = t K H t, sdw s, t, 1 where W t, t, T is a standard Wiener process, K H t, s = c H s 1 H t s u 1 s H 3 u H 1 du if t > s and it is zero otherwise, c H = HH 1 and β H,H 1

7 ESTIMATIO AD REPRODUCTIO FOR HERMITE PROCESSES 167 β, is the Beta function. For t > s, the kernel s derivative is KH t t, s = c s 1 H H t t s H 3. Fortunately we will not need to use these expressions explicitly, since they will be involved below only in integrals whose expressions are known. We will denote by Z q,h t t,1 the Hermite process with self-similarity parameter H 1/, 1. Here q 1 is an integer. The Hermite process can be defined in two ways: as a multiple integral with respect to the standard Wiener process W t t,1 ; or as a multiple integral with respect to a fractional Brownian motion with suitable Hurst parameter. We adopt the first approach throughout the paper. We refer to 1 of 19 for the following integral representation of Hermite processes. t,1 of order q 1 and with self- Definition.1. The Hermite process Z q,h t similarity parameter H 1, 1 is given by t Z q,h t = dh, q 1 K H u, y K H u, y q du y 1... y q dw y1... dw yq,t q.4 for t, 1, where K H is the usual kernel of the fractional Brownian motion and H = 1 + H 1 q H q = H..5 Of fundamental importance is the fact that the covariance of Z q,h is identical to that of fbm, namely E Z q,h = 1 th + s H t s H. Z q,h s t The constant dh, q, given in 1. on page 163, is chosen to avoid any additional multiplicative constants. We stress that Z q,h is far from Gaussian for q > 1, since it is formed of multiple Wiener integrals of order q. The basic properties of the Hermite process are listed below: i the Hermite process Z q,h is H-selfsimilar and it has stationary increments; ii the mean square of the increment is given by Z q,h E Z q,h = t s H ;.6 t s as a consequence, it follows from Kolmogorov s continuity criterion that Z q,h has Hölder-continuous paths of any order δ < H; iii it exhibits long-range dependence in the sense that E Z q,h 1 Z q,h n+1 Z n q,h =. n 1 In fact, the summand in this series is of order n H. This property is identical to that of fbm since the processes share the same covariance structure, and the property is well-known for fbm with H > 1/; iv if q = 1 then Z 1,H is a standard Brownian motion with Hurst parameter H while for q this stochastic process is not Gaussian. In the case q = this stochastic process is known as the Rosenblatt process.

8 168 ALEXADRA CHROOPOULOU, CIPRIA A. TUDOR, AD FREDERI G. VIES 3. Variations of the Hermite Process Since E Z q,h Z q,h i = H and by.6, the centered quadratic variation statistic V given in the introduction can be written as 1 V = H 1 Z q,h i= Z q,h i H. Let := i,. In preparation for calculating the variance of V we will find an explicit expansion of V in Wiener chaos. We have Z q,h Z q,h i = I q f i, where we denoted by f i, y 1,..., y q the expression 1, y 1... y q dh, q 1 K H u, y K H u, y q du y 1... y q i 1, i y 1... y q dh, q 1 K H u, y K H u, y q du. y 1... y q Using the product formula for multiple integrals., we obtain q q I q f i, I q f i, = l! I q l f i, l f i, k l= where the f l g denotes the l-contraction of the functions f and g given by.3. Let us compute these contractions; for l = q we have f i, q f i, = q! f i,, f i, L,1 q = E Z q,h Z q,h i = H. Throughout the paper the notation 1 Kt, s will be used for 1 K H t, s. For l = we have f i, f i, y 1,... y q, z 1,..., z q = f i, f i, y 1,... y q, z 1,..., z q while for 1 k q 1 = f i, y 1,..., y q f i, z 1,..., z q f i, k f i, y 1,... y q k, z 1,..., z q k = dh, q,1 k dα 1... dα k 1 y i,q k 1α i,k 1 y i i,q k 1α i i,k 1 z i,q k 1α i,k 1 z i i,q k 1α i i,k I y,k I y i,k I z,k i I z i,k du 1 Ku, y Ku, y q k 1 Ku, α Ku, α k du 1 Ku, y Ku, y q k 1 Ku, α Ku, α k dv 1 Kv, z Kv, z q k 1 Kv, α Kv, α k dv 1 Kv, z Kv, z q k 1 Kv, α Kv, α k

9 ESTIMATIO AD REPRODUCTIO FOR HERMITE PROCESSES 169 where 1 xj i,k denotes the indicator function 1, i kx j with x being y, z, or α, and I x i,k denotes the interval x 1 x q k α 1 α k ;i/, with x being y or z. By interchanging the order of the integration we get f i, k f i, y 1,... y q k, z 1,..., z q k = dh, q {1, q ky i, z i z 1...z q k dv 1 Kv, z Kv, z q k 1, q ky i1, i q kz i i z 1...z q k dv 1 Kv, z Kv, z q k 1, i q ky i1, q kz i y 1...y q k du 1 Ku, y Ku, y q k u v k 1 Ku, α 1 Kv, αdα y 1...y q k du 1 Ku, y Ku, y q k i z 1...z q k dv 1 Kv, z Kv, z q k +1, i q ky i1, i q kz i i and since i z 1...z q k dv 1 Kv, z Kv, z q k u v with ah = H H 1, we obtain u v k 1 Ku, α 1 Kv, αdα y 1...y q k du 1 Ku, y Ku, y q k u v k 1 Ku, α 1 Kv, αdα y 1...y q k du 1 Ku, y Ku, y q k u v 1 Ku, α 1 Kv, αdα = ah u v H k } 1 Ku, α 1 Kv, αdα f i, k f i, y 1,... y q k, z 1,..., z q k = dh, q ah k { 1, q ky i1, q kz i du 1 Ku, y Ku, y q k y 1...y q k dv 1 Kv, z Kv, z q k u v H k z 1...z q k 1, q ky i1, i q kz i y 1...y q k du 1 Ku, y Ku, y q k i dv 1 Kv, z Kv, z q k u v H k z 1...z q k

10 17 ALEXADRA CHROOPOULOU, CIPRIA A. TUDOR, AD FREDERI G. VIES i 1, i q ky i1, q kz i du 1 Ku, y Ku, y q k y 1...y q k dv 1 Kv, z Kv, z q k u v H k z 1...z q k i +1, i i1, i i du 1 Ku, y Ku, y q k y 1...y q k i } dv 1 Kv, z Kv, z q k u v H k z 1...z q k. As a consequence, we can write 3.1 V = T q + c q T q c 4 T 4 + c T 3. where q c q k := k! k 3.3 are the combinatorial constants from the product formula for k q 1, and 1 T q k := H 1 I q k f i, k f i,, 3.4 where the integrands in the last formula above are given explicitly in 3.1. This Wiener chaos decomposition of V allows us to find V s precise order of magnitude via its variance s asymptotics. Proposition 3.1. With c 1,H = it holds that i= 4dH, q 4 H H 1 q 4H 34H H q H 1q 1 + 1, 3.5 lim E c 1 1,H H c V = 1. Proof. We only need to estimate the L norm of each term appearing in the chaos decomposition 3. of V, since these terms are orthogonal in L. We can write, for k q 1, E Tq k = 4H q k! 1 s f i, k f i, i= L,1 q k 1 = 4H q k! f i, k f i,, f j, k f j, L,1 q k i,j= where g s = g and f i, k f i, denotes the symmetrization of the function f i, k f i,. We will consider first the term T obtained for k = q 1. In this case, the

11 ESTIMATIO AD REPRODUCTIO FOR HERMITE PROCESSES 171 kernel 1 i= f i, q 1 f i, is symmetric and we can avoid its symmetrization. Therefore E T 1 =! 4H f i, q 1 f i, L,1 i= 1 =! 4H f i, q 1 f i,, f j, q 1 f j, L,1. i,j= We compute now the scalar product in the above expression. By using Fubini theorem, we end up with the following easier expression f i, q 1 f i,, f j, q 1 f j, L,1 = ah q dh, q 4 u v H q 1 u v H q 1 u u H v v H dv du dvdu Using the change of variables y = u i and similarly for the other variables, we now obtain E T = dh, q 4 H H 1 q 4H 4 H q 1 i,j= dydzdy dz y z H q 1 y z H q 1 y y + i j H z z + i j H. This can be viewed as the sum of a diagonal part i = j and a non-diagonal part i j, where the non-diagonal part is dominant, as the reader will readily check. Therefore, the behavior of E T will be given by E T :=!dh, q 4 H H 1 q i>j dydzdy dz y z y z H q 1 y y + i j z z + i j H =!dh, q 4 H H 1 q i i= l= I j dydzdy dz y z y z H q 1 y y + l 1 z z + l 1 H =!dh, q 4 H H 1 q l + 1 l= I j dydzdy dz y z y z H q 1 y y + l 1 z z + l 1 H. As in note that 1 l + 1 y y + l 1 H z z + l 1 H l=

12 17 ALEXADRA CHROOPOULOU, CIPRIA A. TUDOR, AD FREDERI G. VIES = H 1 1 l 1 y y + l 1 H z z + l 1 H. l= Using a Riemann sum approximation argument we conclude that E T 4dH, q 4 H H 1 q H 4H 34H H q H 1q Therefore, it follows that E c 1 1,H H T 1, 3.6 with c 1,H as in 3.5. Let us study now the term T 4,..., T q given by 3.4. Here the function 1 i= f i, k f i, is no longer symmetric but we will show that the behavior of its L norm is dominated by E T. Since for any square integrable function g one has g L g L, we have for k =,..., q 1 q k! E 1 Tq k = 4H f i, k f i, L,1 q k i= 1 4H f i, k f i, L,1 q k i= 1 = 4H f i, k f i,, f j, k f j, L,1 q k 3.7 i,j= and proceeding as above, with e H,q,k := q k!h H 1 q dh, q 4 we can write E 1 Tq k eh,q,k 4H dy 1 dz 1 dy 1dz 1 i,j= I j I j y 1 z 1 H k y 1 z 1 H k y 1 y 1 H q k z 1 z 1 H q k and using a change of variables as before, E Tq k 1 e H,q,k 4H 4 H q dydzdy dz y z y z H,1 4 = i,j= k y y + i j H q k z z + i j H q k q k!dh, q4 H q k l 1 ah q 1 dydzdy dz l=,1 4 y z y z H k y y + l 1 z z + l 1 H q k 3.8.

13 ESTIMATIO AD REPRODUCTIO FOR HERMITE PROCESSES 173 Since off a diagonal term again of lower order, the terms z z are dominated by l for large l, it follows that, for 1 k q 1 E c 1 q k,h H q k Tq k = O1 3.9 when, with c q k,h 1 = 1 xx H q k dxah q k!dh, q ah q. 3.1 It is obvious that the dominant term in the decomposition of V is the term in the chaos of order. The case k = is in the same situation for H > 3 4 and for H 1, 3 4 the term T q obtained for k = has to be renormalized by, as in proofs in the remainder of the paper; in any case it is dominated by the term T. More specifically we have for any k q, E H Tq k = O H q k Combining this with the orthogonality of chaos integrals, we immediately get that, up to terms that tend to, H V and H T have the same norm in L Ω. This finishes the proof of the proposition. Summarizing the spirit of the above proof, to understand the behavior of the renormalized sequence V it suffices to study the limit of the term 1 I H 1 H f i, q 1 f i, 3.1 with 1, i y z +1, i y1 I i z i= f i, q 1 f i, y, z 3.13 = dh, q ah q 1 dvdu 1 Ku, y 1 Kv, z u v H z q 1 dvdu 1 Ku, y 1 Kv, z u v H q 1 +1 Ii y1, i z dvdu 1 Ku, y 1 Kv, z u v H q 1 y +1 Ii y1 Ii z dvdu 1 Ku, y 1 Kv, z u v H q 1. y z We will see in the proof of the next theorem that, of the contribution of the four terms on the right-hand side of 3.13, only the first one does not tend to in L Ω. Hence the following notation will be useful: f will denote the integrand

14 174 ALEXADRA CHROOPOULOU, CIPRIA A. TUDOR, AD FREDERI G. VIES of the contribution to 3.1 corresponding to that first term, and r will be the remainder of the integrand in 3.1. In other words, and 1 i= 1 f + r = H 1 H f i, q 1 f i, 3.14 i= f y, z : = H 1 H dh, q ah q 1 1, i y z dvdu 1 Ku, y 1 Kv, z u v H q Theorem 3.. The sequence given by 3.1 converges in L Ω as to the constant c 1/ 1,H times a standard Rosenblatt random variable 1 Z,H 1 with selfsimilarity parameter H 1 and H is given by.5. Consequently, we also have that c 1/ 1,H H c 1 V converges in L Ω as to the same Rosenblatt random variable. Proof. The first statement of the theorem is that H T converges to c 1/ 1 1,H Z,H 1 in L Ω. From 3.1 it follows that T is a second-chaos random variable, with kernel 1 H 1 f i, q 1 f i, i= see expression in 3.13, so we only need to prove this kernel converges in L, 1. The first observation is that r y, z defined in 3.14 converges to zero in L, 1 as. The crucial fact is that the intervals which are disjoints, appear in the expression of this term and this implies that the nondiagonal terms vanish when we take the square norm of the sum; in fact it can easily be seen that the norm in L of r corresponds to the diagonal part in the evaluation in ET which is clearly dominated by the non-diagonal part, so this result comes as no surprise. The proof follows the lines of. This shows H T is the sum of I f and a term which goes to in L Ω. Our next step is thus simply to calculate the limit in L Ω, if any, of I f where f has been defined in By the isometry property.1, limits of second-chaos r.v. s in L Ω are equivalent to limits of their symmetric kernels in L, 1. ote that f is symmetric. Therefore, it is sufficient to prove that f converges to the kernel of the Rosenblatt process at time 1. We have by definition f y, z = H H 1 q 1 dh, q H 1 H 1 u v H q 1 1 K H u, y 1 K H v, z. Thus for every y, z, i= f y, z = dh, q H H 1 q 1 H 1 H

15 ESTIMATIO AD REPRODUCTIO FOR HERMITE PROCESSES i= u v H q 1 1 K H u, y 1 K H v, zdudv 1 = dh, q H H 1 q 1 H 1 H u v H i= 1 K H u, y 1 K H v, z 1 K H i/, z 1 K H i/, z dudv +dh, q H H 1 q 1 H 1 H u v H i= 1 K H i/, y 1 K H i/, zdudv q 1 q 1 =: A 1 y, z + A y, z. As in, one can show that E A 1 as. Regarding the L,1 second term A y, z, the summand is zero if i/ < y z, therefore we get that f is equivalent to 1 H 1 H dh, q H H 1 q 1 u v H q 1 1 K H i/, y 1 K H i/, zdudv = H H 1 q 1 dh, q H 1 H 1 i= 1 K H i/, y 1 K H i/, z1 y z<i/ i= u v H q 1 dudv = H H 1 q 1 H q 1 + 1H 1q = H 1 H q 1 i= 1 K H i/, y 1 K H i/, z1 y z<i/ dh, q H H 1 q 1 dh, H q 1 + 1H 1q dh, 1 i= 1 K H i/, y 1 K H i/, z1 y z<i/. The sequence dh, 1 1 i= 1K H i/, y 1 K H i/, z1 y z<i/ is a Riemann sum that converges pointwise on, 1 to the kernel of the Rosenblatt process Z H 1, at time 1. To obtain the convergence in L, 1 we will apply the dominated convergence theorem. Indeed, i= 1 K H i/, y 1 K H i/, z1 y z<i/ dydz

16 176 ALEXADRA CHROOPOULOU, CIPRIA A. TUDOR, AD FREDERI G. VIES = 1 1 i,j= 1 1 K H i/, y 1 K H j/, y1 y<i j/ dy 1 1 E Z i/ Z j/, i,j= where Z i/ is the difference Z i Z i 1 for a Rosenblatt process Z. We now show that the above sum is always, which proves that the last expression, with the factor, is bounded. In fact for H 1 = H 1 = 1 i,j= 1 i,j= = 4H 1 4 E Z i/ Z j/ i j i,j= 4H 1 4 H1 + H i j 1 i j 1 H1 i j + 1 H1 + i j 1 H1 i j H1 1 l= +1 l + 1 H1 + l 1 H1 l H1. The function gl = l+1 H 1 + l 1 H 1 l H 1 behaves like H 1 H 1 1 l H 1 for large l. We need to separate the cases of convergence and divergence of the series gl. It is divergent as soon as H 1 3/4, or equivalently H 7/8, in which case we get for some constant c not dependent on, 1 i,j= E Z i/ Z j/ c 4H H 1 3 = c. The series gl is convergent if H < 7/8, in which case we get 1 i,j= E Z i/ Z j/ c 4H For this to be, we simply need 4H <, i.e. H > 5/8. However, since q and H > 1/ we always have H > 3/4. Therefore in all cases, the sequence A y, z is bounded in L, 1 and in this way we obtain the L convergence to the kernel of a Rosenblatt process of order 1. The first statement of the theorem is proved. In order to show that c 1/ 1,H H c 1 V converges in L Ω to the same Rosenblatt random variable as the normalized version of the quantity in 3.1, it is sufficient to show that, after normalization by H, each of the remaining terms of in the chaos expansion 3. of V, converge to zero in L Ω, i.e. that H T q k converge to zero in L Ω, for all 1 k < q 1. From

17 ESTIMATIO AD REPRODUCTIO FOR HERMITE PROCESSES we have E H Tq k = O H q k 1 which is all that is needed, concluding the proof of the theorem. 4. Reproduction Property for the Hermite Process We now study the limits of the other terms in the chaos expansion 3. of V. We will consider first the convergence of the term of greatest order T q in this expansion. The behavior of T q is interesting because it differs from the behavior of the all other terms: its suitable normalization possesses a Gaussian limit if H 1/, 3/4. Therefore it inherits, in some sense, the properties of the quadratic variations for the fractional Brownian motion results in. We have 1 T q = H 1 I q f i, f i, and E Tq = 4H q! 1 i,j= f i, f i,, f j, f j, L,1 q. We will use the following combinatorial formula: if f, g are two symmetric functions in L, 1 q, q! f f, g g L,1 q q 1 q = q! f f, g g L,1 q + q! f k g, g k f k L,1q k k=1 i= to obtain E 1 Tq = 4H q! f i,, f j, L,1 q First note that E i,j= 1 + 4H q! q 1 q f i, k f j,, f j, k f i, k L,1q k. Z q,h j = E I q f i, I q f j, = i,j= k=1 Z q,h Z q,h i Z q,h j+1 q! f i,, f j, L,1 q and so the covariance structure of Z q,h implies = 1 4 q! 1 1 i,j= f i,, f j, L,1 q i,j= i H H H j + 1 i j 1 i j Secondly, the square norm of the contraction f i, k f j, has been computed before actually its expression is obtained in the lines from formula 3.7 to formula 3.8. By a simple polarization, we obtain H 1 1 f i, k f j,, f j, k f i, L,1 q k i,j=

18 178 ALEXADRA CHROOPOULOU, CIPRIA A. TUDOR, AD FREDERI G. VIES = dh, q 4 ah q 4H H q 1 i,j= y z + i j y z + i j H k y y + i j z z + i j H q k dydzdy dz and as in the proof of Proposition 3.1 we can find that this term has to be renormalized by, if H > 3 4 b H,k H q = b H,k 4 4H where b 1,H,k = q! C q k dh, q 4 ah q 1 1 xx4h 4 dx. If H 1, 3 4, then, the same quantity will be renormalized by b,h,k where b,h,k = q! C q k dh, q 4 ah q k=1 k H k + 1 H k 1 H while for H = 3 4 the renormalization is of order b 3,H,klog 1 with b 3,H,k = q! C q k dh, q 4 ah q 1/. As a consequence, we find a sum whose behavior is well-known it is the same as the mean square of the quadratic variations of the fractional Brownian motion, see e.g. and we get, for large E Tq 1 x 1,H, if H 1, 3 4 ; E Tq 4H 4 x,h, if H 3 4, 1 E Tq log x 3,H, if H = 3 4 q 1 where x 1,H = l=1 b,h,l / k=1 k H k + 1 H k 1 H, x,h = q 1 l=1 b 1,H,l + H H 1/4H 3 and x 3,H = q 1 l=1 b 3,H,l + 9/16. Remark 4.1. The fact that the normalizing factor for T q when H < 3/4 is 1/ in particular does not depend on H is a tell-tale sign that normal convergence may occur. It is possible to show that the term in the chaos of order q converges, after its renormalization, to a Gaussian law. More precisely: a suppose that H 1, 3 4 and let F := x 1/ 1,H Tq ; then, as, the sequence F convergence to the standard normal distribution, 1; b suppose H = 3 4 and set G := log x 1/ 3,H T q; then, as, the sequence G convergence to the standard normal distribution, 1. We refer to the extended version of our paper on arxiv for the basic ideas of the proof. It is possible to give the limits of the terms T q to T appearing in the decomposition of the statistics V. All these renormalized terms will converge to Hermite random variables of the same order as their indices we have already proved this

19 ESTIMATIO AD REPRODUCTIO FOR HERMITE PROCESSES 179 property in detail for T in the previous section. This is a kind of reproduction of the Hermite processes through their variations. Theorem 4.. For every H 1, 1 and for every k = 1,..., q we have lim H q k T q k = z k,h Z q k,q kh 1+1, in L Ω 4. q k,q kh 1+1 where Z denotes a Hermite random variable with self-similarity parameter q kh and z k,h = dh, q ah k H 1k H Moreover, if H 3 4, 1 then lim H x 1/,H T q = Z q,h 1, in L Ω. 4.3 Proof. Recall that we have T k = H 1 I q k 1 i= f i, k f i,, for k = 1 to q, with f i, k f i, given by 3.1. The first step of the proof is to observe that the limit of H q k T q k is given by H q k H 1 I q k f q k with fq k y 1,..., y q k, z 1,..., z q k = dh, q ah k 1 i= 1, i y i1, i z i 1 Ku, y Ku, y q k 1 Kv, z Kv, z q k u v H k dvdu. The argument leading to the above fact is the same as in Theorem 3.: a the remainder term r q k, which is defined by T q k minus I fq k converges to zero similarly to the term r in the proof of Theorem 3. because of the appearance of the indicator functions 1 Ii y i or 1 Ii z i in each of the terms that form this remainder. The second step of the proof is to replace 1 Ku, y i by 1 K i, y i and 1 Kv, z i by 1 K i, z i on the interval inside the integrals du and dv. This can be argued, as in the proof of Theorem 3., by a dominated convergence theorem. Therefore the term H q k T q k will have the same limit as = 1 H q k H 1 dh, q ah k q k j=1 i= 1, i y i1, i z i 1 K i, y j 1 K i, z j u v H k dvdu Ii dh, q ah k H 1k + 1 H H q k H 1 H k+ 1 i= = z k,h 1, i y i, z i 1 i= q k j=1 1, i y i, z i 1 K i, y j 1 K i, z j q k j=1 1 K i, y j 1 K i, z j.

20 18 ALEXADRA CHROOPOULOU, CIPRIA A. TUDOR, AD FREDERI G. VIES ow, the last sum is a Riemann sum that converges pointwise and in L, 1 q k to z k,h L 1 where L 1 y 1,..., y q k, z 1,..., z q k = 1 q k y 1 z q k j=1 1 Ku, y j 1 Ku, z j du which is the kernel of the Hermite random variable of order q k with selfsimilarity parameter q kh The case H 3 4, 1 and k = follows analogously. 5. Consistent Estimation of H and its Asymptotics Theorem 3. can be immediately applied to the statistical estimation of H. We note that with V in 1.1 and S defined by S = 1 1 i= Z q Z q i, 5.1 we have 1 + V = H S 5. and E S = H so that H = log ES log. To form an estimator of H, since Theorem 3. implies that S evidently concentrates around its mean, we will use S in the role of an empirical mean, instead of its true mean; in other words we let Ĥ = Ĥ = log S log. We immediately get from 5. that log 1 + V = H log + log S = H Ĥ log. 5.3 The first observation is that Ĥ is strongly consistent for the Hurst parameter. Proposition 5.1. We have that Ĥ converges to H almost surely as. Proof. Let us prove that V converges to zero almost surely as. We know that V converges to in L Ω as and an estimation for its variance is given by the formula 3.6. On the other hand this sequence is stationary because the increments of the Hermite process are stationary. We can therefore apply a standard argument to obtain the almost sure convergence for discrete stationary sequence under condition 3.6. This argument follows from Theorem 6., page 49 in 6 and it can be used exactly as in the proof of Proposition 1 in 4. A direct elementary proof using the Borel-Cantelli lemma is almost as easy. The almost sure convergence of Ĥ to H is then obtained immediately via 5.3. Owing to the fact that V is of small magnitude it converges to zero almost surely by the last proposition s proof, log 1 + V can be confused with V. It then stands to reason that H Ĥ H log is asymptotically Rosenblattdistributed since the same holds for V by Theorem 3.. Just as in that theorem, more is true: as we now show, the convergence to a Rosenblatt random variable occurs in L Ω.

21 ESTIMATIO AD REPRODUCTIO FOR HERMITE PROCESSES 181 Proposition 5.. There is a standard Rosenblatt random variable R with selfsimilarity parameter H 1 such that Ĥ lim E H H log c c 1/ 1,H,q R = Proof. To simplify the notation, we denote c c 1/ 1,H,q by c in this proof. Theorem 3. signifies that a standard Rosenblatt r.v. R with parameter H 1 exists such that lim E H V cr =. From 5.3 we immediately get H E H Ĥ log cr H = E log 1 + V cr H E V cr + 4 4H E V log 1 + V. Therefore, we only need to show that E V log 1 + V = o 4H 4. Using the inequality x log1 + x = x log1 x x for x 1 E V log1 + V one gets E V log1 + V 1 V 1 + E V log1 + V 1 V < 1 EV 4 + 4P V < 1 + 4E log1 + V 1 V < 1. The first term is bounded above and this is immediately dealt with using the following lemma. Lemma 5.3. For every n, there is a constant c n such that E V n c n 4H 4n. Proof. In the proof of Proposition 3.1 we calculated the L norm of the terms appearing in the decomposition of V, where V is a sum of multiple integrals. Therefore, from Proposition 5.1 of 1 we immediately get an estimate for any event moment of each term. Indeed, for k = q 1 E T n and for 1 k < q 1 E T n q k n 1 c 1,H 4H 4 n = cn 4H 4n q kn c 1,q,H q kh n = c q,n q kh n The dominant term is again the T term and the result follows immediately. To continue the proof of Proposition 5., apply the above lemma for n = and use a similar proof to prove that E log1 + V 1 V < 1 = o 4H 4 and P V < 1 = O 8H 8 = o 4H 4. This leads to Proposition 5.

22 18 ALEXADRA CHROOPOULOU, CIPRIA A. TUDOR, AD FREDERI G. VIES A difficulty arises when applying the above proposition for model validation when checking the asymptotic distribution of the estimator Ĥ: the normalization constant H log depends on H. While it is not always obvious that one may replace this instance of H by its estimator, in our situation, because of the L Ω convergences, this is legitimate, as the following theorem shows. Theorem 5.4. Let Ĥ = 1 + Ĥ 1/q. There is a standard Rosenblatt random variable R with self-similarity parameter H 1 such that lim E Ĥ H Ĥ log c c 1/ 1,H,q R =. Proof. By the previous proposition, it is sufficient to prove that lim E Ĥ H H Ĥ log =. We decompose the probability space depending on whether Ĥ is far or not from its mean. For a fixed value ε > which will be chosen later, it is most convenient to define the event } D = {Ĥ > qε/ + H 1. We have E Ĥ H H Ĥ log = E 1 D Ĥ H H Ĥ log +E 1 D c Ĥ H H Ĥ log =: A + B. We study A first. Introducing the shorthand notation x = max H, Ĥ and y = min H, Ĥ, we may write Ĥ H = e x log e y log = e y log e x y log 1 y log x y x y = log x H Ĥ = q 1 log x H Ĥ. Thus A q 1 E 1 D x H Ĥ log = q 1 E x 4 4H 1D 4 4H H Ĥ log. ow choose ε, H. In this case, if ω D, and x = H, we get x 4 4H = x < ε. On the other hand, for ω D and x = Ĥ, we get x 4 4H = Ĥ 4 4H < ε as well. In conclusion, on D, x 4 4H < ε,

23 ESTIMATIO AD REPRODUCTIO FOR HERMITE PROCESSES 183 which implies immediately A ε q 1 E 4 4H H Ĥ log ; this prove that A tends to as, since the last expectation above is bounded converges to a constant by Proposition 5.. ow we may study B. We are now operating with ω D c. In other words, H Ĥ > 1 H qε/. Still using ε < H, this implies that H > Ĥ. Consequently, it is not inefficient to bound H Ĥ above by Ĥ. In the same fashion, we bound H Ĥ above by H. Hence we have, using Hölder s inequality with the powers q and p 1 + q 1 = 1. B H log E 1 D c Ĥ 5.4 H log P 1/p D c E 1/q Ĥ q. From Proposition 5., by Chebyshev s inequality, we have H E 1/p Ĥ P 1/p D c 1 H qε/ c q,h 4 4H /p /p 5.5 for some constant c q,h depending only on q and H. Dealing with the other term in the upper bound for B is a little less obvious. We must return to the definition of Ĥ. By 5.3 we have Therefore, 1 + V = H Ĥ = qh Ĥ = q Ĥ q H. E 1/q Ĥ q H E 1/q 1 + V Plugging 5.5 and 5.6 back into 5.4, we get H. 5.6 B Hc q,h log 4 4H /p 1. Given that q 4 and p is conjugate to q, we have p 4/3 so that /p 1 1/ >, which proves that B goes to as. This finishes the proof of the theorem. Finally we state the extension of our results to L p Ω-convergence. Theorem 5.5. The convergence in Theorem 5.4 holds in any L p Ω. In fact, the L Ω-convergences in all other results in this paper can be replaced by L p Ω- convergences.

24 184 ALEXADRA CHROOPOULOU, CIPRIA A. TUDOR, AD FREDERI G. VIES Proof. We only give the outline of the proof. Lemma 5.3 works because, in analogy to the Gaussian case, for random variables in a fixed Wiener chaos of order p, existence of second moments implies existence of all moments, and the relations between the various moments are given using a set of constants which depend only on p. This Lemma can be used immediately to prove the extension of Proposition 3.1 that E V p c p p4h 4. Proving a new version of Theorem 3. with L p Ω-convergence can base itself on the above result, and requires a careful reevaluation of the various terms involved. Proposition 5. can then be extended to L p Ω-convergence thanks to the new L p Ω versions of Theorem 3. and Proposition 3.1, and Theorem 5.4 follows easily from this new version of Proposition 3.1. References 1. Beran, J.: Statistics for Long-Memory Processes, Chapman and Hall, Breuer, P. and Major, P.: Central limit theorems for nonlinear functionals of Gaussian fields, J. Multivariate Analysis, 13, Burdzy, K. and Swanson, J.: Variation for the solution to the stochastic heat equation II, Preprint, Coeurjolly, J. F.: Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths, Statistical Inference for Stochastic Processes Dobrushin, R. L. and Major, P.: on-central limit theorems for non-linear functionals of Gaussian fields, Z. Wahrscheinlichkeitstheorie verw. Gebiete Doob, J. L.: Stochastic Processes. Wiley Classics Library, Giraitis, L. and Surgailis, D.: CLT and other limit theorems for functionals of Gaussian processes, Z. Wahrsch. verw. Gebiete Guyon, X. and León, J.: Convergence en loi des H-variations d un processus gaussien stationnaire sur R, Annales IHP Istas, J. and Lang, G.: Quadratic variations and estimators of the Hölder index of a Gaussian process, Annales IHP Major, P.: Tail behavior of multiple random integrals and U-statistics, Probability Surveys, ourdin, I.: Asymptotic behavior of certain weighted quadratic variations and cubic variations of fractional Brownian motion, The Annals of Probability, ourdin,i., ualart, D. and Tudor, C. A.: Central and non-central limit theorems for weighted power variations of fractional Brownian motion, Annales IHP 7 to appear. 13. ourdin, I. and Peccati, G.: Weighted power variations of iterated Brownian motion, Electronic Journal of Probability, 13, ualart, D.: Malliavin Calculus and Related Topics, Second Edition, Springer, Réveillac, A.: Convergence of finite-dimensional laws of the weighted quadratic variations process for some fractional Brownian sheets, Stoch. Anal. Appl., 7, Swanson, J.: Variations of the solution to a stochastic heat equation, Annals of Probability, 35, Taqqu, M.: Weak convergence to the fractional Brownian motion and to the Rosenblatt process, Z. Wahrscheinlichkeitstheorie verw. Gebiete, Taqqu, M.: Convergence of integrated processes of arbitrary Hermite rank, Z. Wahrscheinlichkeitstheorie verw. Gebiete, Tudor, C. A.: Analysis of the Rosenblatt process, ESAIM Probability and Statistics, Tudor, C. A. and Viens, F.: Variations and estimators for the selfsimilarity order through Malliavin calculus, The Annals of Probability,

25 ESTIMATIO AD REPRODUCTIO FOR HERMITE PROCESSES 185 Alexandra Chronopoulou: Dept. Statistics, Purdue University, West Lafayette, Indiana I 4796, USA address: achronop@stat.purdue.edu Ciprian A. Tudor: Laboratoire Paul Painlevé, Université de Lille 1, F Villeneuve d Ascq, France address: tudor@math.univ-lille1.fr Frederi G. Viens: Department of Statistics, Purdue University, West Lafayette, Indiana I 4796, USA address: viens@stat.purdue.edu