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Wiener integrals with respect to the Hermite process

and a Non-Central Limit Theorem

Makoto Maejima, Ciprian Tudor

To cite this version:

Makoto Maejima, Ciprian Tudor. Wiener integrals with respect to the Hermite process and a Non-Central Limit Theorem. Stochastic Analysis and Applications, Taylor & Francis: STM, Behavioural Science and Public Health Titles, 2007, 25 (5), pp.1043 - 1056. �10.1080/07362990701540519�. �hal-00176227�

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Wiener integrals with respect to the Hermite process and a

Non-Central Limit Theorem

Makoto Maejima 1 Ciprian A. Tudor 2∗ 1Department of Mathematics, Keio University ,

3-14-1, Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan. maejima@math.keio.ac.jp

2 SAMOS-MATISSE, Centre d’Economie de La Sorbonne,

Universit´e de Panth´eon-Sorbonne Paris 1, 90, rue de Tolbiac, 75634 Paris Cedex 13, France.

tudor@univ-paris1.fr

February 2, 2007

Abstract

We introduce Wiener integrals with respect to the Hermite process and we prove a Non-Central Limit Theorem in which this integral appears as limit. As an example, we study a generalization of the fractional Ornstein-Uhlenbeck process.

2000 AMS Classification Numbers: 60F05, 60H05, 60G18.

Key words: Hermite process, fractional Brownian motion, non-central limit theo-rems, Wiener integral, Ornstein-Uhlenbeck process.

1

Introduction

The selfsimilar processes have been widely studied due to their applications as models for various phenomena, like hydrology, network traffic analysis and mathematical finance.

An interesting class of selfsimilar processes is given as limits of the so called Non-Central Limit Theorem studied in [21] and [9]. We briefly recall the context. Let g be a function of Hermite rank k (see Section 5 for the definition) and let (ξn)n∈Z be a stationary

Gaussian sequence with mean 0 and variance 1 which exhibits long range dependence in the sense that the correlation function satisfies

r(n) := E (ξ0ξn) = n

2H−2 k L(n) ∗Research supported by the Japan Society for the Promotion of Science.

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where H ∈ (12,1), k ≥ 1 and L is a slowly varying function at infinity (see e.g. [10] for the definition). Then the following holds:

1 nH [nt] X j=1 g(ξj) (1)

converges as n → ∞ in the sense of finite dimensional distributions, to the process ZHk(t) = c(H, k) Z Rk ÃZ t 0 Ã k Y i=1 (s − yi) −(1 2+1−Hk ) + ! ds ! dB(y1) . . . dB(yk) (2)

where the above integral is a Wiener-Itˆo multiple integral of order k with respect to the standard Brownian motion (B(y))y∈R and c(H, k) is a positive normalization constant depending only

on H and k. The process (Zk

H(t))t≥0 is called the Hermite process and it is H-selfsimilar in

the sense that for any c > 0, (Zk

H(ct)) =(d) (cHZHk(t)), where ” =(d) ” means equivalence of all

finite dimensional distributions, and it has stationary increments.

The most studied Hermite process is of course the fractional Brownian motion (which is obtained in (2) for k = 1) due to its large range of applications. Recently, a rich theory of stochastic integration with respect to this process has been introduced and stochastic differen-tial equations driven by the fractional Brownian motion have been considered. We refer to [4], [8] or [11], to cite only a few. The process obtained in (2) for k = 2 is known as the Rosenblatt process. It was introduced by Rosenblatt in [19] and it has been called in this way by M. Taqqu in [20]. The Rosenblatt process has also practical applications and different aspects of this process, like wavelet type expansion or extremal properties have been studied in [1], [2], [15] or [16].

The aim of this paper is to make a first step in the direction of a stochastic calculus driven by the Hermite process of order k by introducing Wiener integrals with respect to this process. The basic observation is the fact that the covariance structure of the Hermite process is similar to the one of the fractional Brownian motion and this allows the use of the same classes of deterministic integrands as in the fractional Brownian motion case whose properties are known. We will distinguish as in [18], [17], a time domain and a spectral domain of deterministic integrands. As an application, we discuss the existence and the properties of the Hermite Ornstein-Uhlenbeck process (the correspondent of the fractional Ornstein-Uhlenbeck process, see [6]) that appears as the unique solution of a Langevin type equation.

Of central interest is for us a Non-Central Limit Theorem that has as limit the Wiener integral with respect to the Hermite process (Zk

H(t))t≥0 in (2). Recall that it has been proved

in [17] that, if f is a deterministic function in a suitable space, then the sequence 1 nH X j∈Z f µj n ¶ Xj, (3)

where (Xj)j∈Z is in the domain of attraction of the fractional Brownian motion, converges

weakly, as n → ∞, to the Wiener integral RRf(u)dBH(u), where BH denotes the fractional

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the domain of attraction of a stable law, has been studied in [12] and [13]. A natural extension of the convergence of sequences (1) and (3) is to show that

1 nH X j∈Z f µj n ¶ g(ξj)

converges weakly when n → ∞, to the Wiener integral RRf(u)dZHk(u) assuming that the

sequence g(ξj), j ∈ Z belongs to the domain of attraction of the Hermite process (see Theorem

3.4.1 in [10] for such processes or also [7]). We study the time domain approach in which we prove the general result. The spectral domain approach has been considered in [17]. We also mention that in order to not overcharge the paper, we decided to omit the slowly varying function L; indeed, although from the mathematical point of view this function could be sometimes tedious to treat, from the philosophical point of view this is less significant.

We organize our paper as follows. Section 2 recalls several properties of the Hermite process. In Section 3 we construct Wiener integrals with respect to this process and in Section 4 we discuss, as an example, the Hermite Ornstein-Uhlenbechk process which is a generalization of the fractional Ornstein-Uhlenbeck process. Section 5 contains a Non-Central Limit Theorem which has as limit the Wiener integral with respect to the Hermite process.

2

The Hermite process

We will present in this section some basic properties of the Hermite process ¡Zk H(t)

¢

t∈R of

order k ≥ 1, k ∈ Z and with Hurst parameter H ∈ (12,1). This stochastic process is defined as a multiple Wiener-Itˆo integral of order k with respect to the standard Brownian motion B((t))t∈R ZHk(t) = c(H, k) Z Rk Z t 0   k Y j=1 (s − yi) −(1 2+1−Hk ) +   dsdB(y1) . . . dB(yk), (4)

where x+ = max(x, 0). We refer to e.g. [14] for the properties of the multiple stochastic

integrals. For k = 1 the above process is nothing else that the fractional Brownian motion with Hurst parameter H ∈ (0, 1). For k ≥ 2 the process Zk

H is not Gaussian and if k = 2 it is

known (see [20]) as the Rosenblatt process.

Let us compute the covariance R(t, s) := E£Zk

H(t)ZHk(s)

¤

of the Hermite process. By Fubini and the isometry of multiple Wiener-Itˆo integrals, one has

R(t, s) = 2c(H, k)2 Z Rk   Z t 0 Z s 0 k Y j=1 (u − yi) −(1 2+1−Hk ) + (v − yi) −(1 2+1−Hk ) + dvdu   dy1. . . dyk = 2c(H, k)2 Z t 0 Z s 0 Z Rk   k Y j=1 (u − yi) −(1 2+1−Hk ) + (v − yi) −(1 2+1−Hk ) + dy1. . . dyk   dvdu = 2c(H, k)2 Z t 0 Z s 0 ·Z R (u − y)−( 1 2+1−Hk ) + (v − y) −(12+1−Hk ) + dy ¸k dvdu.

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Let β(p, q) =R01zp−1(1 − z)p−1dz, p, q >0 be the beta function. By using the identity Z

R

(u − y)a−1+ (v − y)a−1+ dy= β(a, 2a − 1)|u − v|2a−1

we get R(t, s) = 2c(H, k)2β µ1 2 − 1 − H k , 2H − 2 k ¶kZ t 0 Z s 0 ³ |u − v|2H−2k ´k dvdu = 2c(H, k)2β ¡1 2− 1−Hk , 2H−2 k ¢k H(2H − 1) 1 2(t 2H+ s2H− |t − s|2H).

In order to obtain E(Zk

H(t))2 = 1, we will choose c(H, k)2 = Ã β¡12 −1−Hk ,2H−2k ¢k 2H(2H − 1) !−1

and we will have

R(t, s) = 1 2(t

2H+ s2H− |t − s|2H).

We also recall the following properties of the Hermite process: • The process Zk

H is H-selfsimilar with stationary increments and all moments are finite.

• From the stationarity of increments and the selfsimilarity, it follows that, for any p ≥ 1 Eh¯¯¯ZHk(t) − ZHk(s)¯¯¯pi= c(p, H, k)|t − s|pH.

As a consequence the Hermite process has H¨older continuous paths of order δ < H. We mention that different expressions of the exponent in (4) are used in the literature, but we choose this one in order to have the order of similarity equal to H.

3

Wiener integrals with respect to the Hermite process

In this paragraph we introduce Wiener integrals with respect to the Hermite process. Let us denote by E the class of elementary functions on R of the form

f(u) =

n

X

l=1

al1(tl,tl+1](u), tl< tl+1, al ∈ R, l = 1, . . . , n. (5)

For f ∈ E as above it is natural to define its Wiener integral with respect to the Hermite process Zk H by Z R f(u)dZHk(u) = n X l=1 al ³ ZHk(tl+1) − ZHk(tl) ´ . (6)

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In order to extend the definition (6) to a larger class of integrands, let us make first some observations. By formula (4) we can write

ZHk(t) = Z Rk I¡1[0,t] ¢ (y1, . . . , yk)dB(y1) . . . dB(yk) (7)

where by I we denote the mapping on the set of functions f : R → R to the set of functions f : Rk→ R I(f )(y1, . . . , yk) = c(H, k) Z R f(u) k Y j=1 (u − yi) −(12+1−Hk ) + du.

Note that for k = 1 this operator can be expressed in terms of fractional integrals and deriva-tives (see [18], [3]). Thus the definition (6) can be also written in the form, due to the obvious linearity of I Z R f(u)dZk H(u) = n X l=1 al ³ ZHk(tl+1) − ZHk(tl) ´ = n X l=1 al Z Rk

I(1(tl,tl+1])(y1, . . . , yk)dB(y1) . . . dB(yk)

= Z

Rk

I(f )(y1, . . . , yk)dB(y1) . . . dB(yk). (8)

We introduce now the following space H = {f : R → R;

Z

Rk

(I(f )(y1, . . . , yk))2dy1. . . dyk <∞}

endowed with the norm

kf k2H = Z Rk (I(f )(y1, . . . , yk))2dy1. . . dyk. It holds that kf k2H = c(H, k)2 Z Rk   Z R Z R f(u)f (v) k Y j=1 (u − yi) −(1 2+1−Hk ) + (v − yi) −(1 2+1−Hk ) + dvdu   dy1. . . dyk = c(H, k)2 Z R Z R f(u)f (v) µZ R (u − y)−( 1 2+1−Hk ) + (v − y) −(1 2+1−Hk ) + dy ¶k dvdu = H(2H − 1) Z R Z R

f(u)f (v)|u − v|2H−2dvdu.

Hence we have H = {f : R → R | Z R Z R

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and kf k2H= H(2H − 1) Z R Z R

f(u)f (v)|u − v|2H−2dvdu. (9) Let us observe that the mapping

f 7→ Z

R

f(u)dZHk(u) (10)

provides an isometry from E to L2(Ω). Indeed, for f of the form (5), it holds that

E£I(f )2¤ = n−1 X i,j=0 aiajE[ZH(ti+1) − ZH(ti)) (ZH(tj+1) − ZH(tj)] = n−1 X i,j=0 aiaj(R(ti+1, tj+1) − R(ti+1, tj) − R(ti, tj+1) + R(ti, tj)) = n−1 X i,j=0 aiajH(2H − 1) Z ti+1 ti Z tj+1 tj |u − v|2H−2dvdu = n−1 X i,j=0 aiajh1(ti,ti+1],1(tj,tj+1]iH= kf k 2 H .

On the other hand, it has been proved in [18] that the set of elementary functions E is dense in H. As a consequence the mapping (10) can be extended to an isometry from H to L2(Ω) and relation (8) still holds.

The followings facts has also been proved in [18]:

• The elements of H may be not functions but distributions; it is therefore more practical to work with subspaces of H that are sets of functions. Such a subspace is

|H| = {f : R → R| Z

R

Z

R

|f (u)||f (v)||u − v|2H−2dvdu <∞}.

Then |H| is a strict subspace of H and we actually have the inclusions

L2(R) ∩ L1(R) ⊂ LH1(R) ⊂ |H| ⊂ H. (11)

• The space |H| is not complete with respect to the norm k · kH but it is a Banach space

with respect to the norm kf k2|H|=

Z

R

Z

R

|f (u)||f (v)||u − v|2H−2dvdu.

• A ”spectral domain” included in H can also be defined as b H = {f ∈ L2(R)| Z R ¯ ¯ ¯ bf(x)¯¯¯2|x|−2H+1dx <∞}, (12)

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where bf denotes the Fourier transform of f . We have again that bH is a strict subspace of H and the inclusion

L2(R) ∩ L1(R) ⊂ LH1(R) ⊂ bH ⊂ H is true. We define kf k2b H= Z R ¯ ¯ ¯ bf(x)¯¯¯2|x|−2H+1dx.

• There are elements in |H| that are not in bH and viceversa.

4

Hermite Ornstein-Uhlenbeck processes

Introducing Wiener integrals with respect to the Hermite process allows us to consider a first example of stochastic differential equation with the Hermite process as driving noise. We will consider the Langevin type stochastic equation

Xt= ξ − λ

Z t

0

Xsds+ σZHk(t), t≥ 0, (13)

where σ, λ > 0, the Hurst parameter H belongs to (12,1) and k ≥ 1 is an integer. The initial condition ξ is a random variable in L0(Ω). The case k = 1 has been considered in [6]. Recall that in this case (when Zk

H = BH, a fractional Brownian motion) the unique solution of

the equation (13), that is understood in a Riemann-Stieltjes sense, has the Wiener integral representation YHξ(t) = e−λt µ ξ+ σ Z t 0 eλudBH(u) ¶ , t≥ 0. When the initial condition is ξ = σR−∞0 eλudB

H(u), the solution of (13) can be written as

YH(t) = σ

Z t −∞

e−λ(t−u)dBH(u) (14)

and it is called the stationary fractional Ornstein-Uhlenbeck process. It follows from [6] that the process (14) is a stationary centered Gaussian process, H-selfsimilar and it has stationary increments. Moreover, it is ergodic and exhibits long range dependence for H ∈ (12,1). Its covariance function behaves as, when t ∈ R, N = 1, 2, ... and s → ∞,

E[YH(t)YH(t + s)] = 1 2σ 2 N X m=1 λ−2n   2n−1Y j=0 (2H − j)   s2H−2n+ O(s2H−2N −2). (15)

When H = 12 the process Y1

2 can be also defined by using the Lamperti transform YH(t) = e−λtBH µ αexp(λ Ht) ¶ (16) and both definitions coincide. When H 6= 12, processes obtained by (14) and (16) are different. The above context can be easily generalized to the case of the Hermite process. More precisely, we have the following result.

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Proposition 1 Let (Zk

H(t))t∈R be a Hermite process of order k and let ξ ∈ L0(R). The

following are true for almost all ω and for every λ, σ > 0.

a. For all t > a, the integralRateλudZHk(u, ω) exists in the Riemann-Stieljes sense and it is equal to

eλtZHk(t, ω) − eλaZHk(a, ω) − λ Z t

a

ZHk(u, ω)eλudu.

Moreover, the function t →RateλudZk

H(u, ω) is continuous.

b. The unique continuous solution of the equation y(t) = ξ(ω) − λ Z t 0 y(s)ds + σZHk(t, ω), t≥ 0 is given by y(t) = e−λt µ ξ(ω) + σ Z t 0 eλudZHk(u, ω) ¶ , t≥ 0. In particular, if ξ = σR−∞0 eλudZHk(u, ω), then

y(t) = σ Z t

−∞

e−λ(t−u)dZHk(u, ω), t≥ 0.

Proof: The proof of Proposition A.1 in [6] and the fact that the process Zk

H is H¨older

continuous of order δ < H imply the conclusion.

As a consequence, the unique solution of (13) with initial condition σR−∞0 eλudZHk(u) is given by

YHk(t) = σ Z t

−∞

e−λ(t−u)dZHk(u), t≥ 0. (17) The process given by (17) will be called Hermite Ornstein-Uhlenbeck process of order k. Be-cause the covariance structure of the Hermite process is the same as the fractional Brownian motion’s one, it is easy to see that the above integral coincides with the Wiener integral defined in Section 3. Hence the process Yk

H is a centered Gaussian process with covariance

EhYHk(t)YHk(s)i= σ2 Z t −∞ Z s −∞ e−λ(t−u)e−λ(s−v)|u − v|2H−2dvdu <∞.

Clearly, relation (15) still holds for the Hermite Ornstein-Uhlenbeck process.

5

Non-Central Limit Theorem

In this section we extend the results in [9] and [21] by proving a Non-Central Limit Theorem which has as limit the Wiener integral with respect to the Hermite process introduced in Section 2.

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We will consider a sequence of centered stationary Gaussian random variables (ξj)j∈Z

with Eξ2

j = 1 and the correlation function

r(n) := E(ξ0ξn) = n

2H−2

k . (18)

Let us recall the notion of Hermite rank. Denote by Hm(x) the Hermite polynomial

of degree m given by Hm(x) = (−1)me x2 2 d m dxme− x2

2 and let g be a function on R such that E[g(ξ0)] = 0 and E£g(ξ0)2¤ < ∞. Assume that g has the following expansion in Hermite

polynomials g(x) = ∞ X j=0 cjHj(x)

where cj = j!1E[g(ξ0)Hj(ξ0)]. The Hermite rank of g is defined by

k= min{j; cj 6= 0}.

Since E [g(ξ0)] = 0, we have k ≥ 1.

We also introduce the sequence of stochastic processes ZHk,n given by

ZHk,n(u) = 1 nH [nu] X l=1 g(ξl) (u ≥ 0) and ZHk,n(u) = 1 nH 0 X j=−[nu]−1 g(ξl) (u < 0), (19)

where g is a function of Hermite rank k. By the results in [9] and [21] (see also Theorem 3.4.1 in [10]), it holds that

ZHk,n →(d) ckZHk, n→ ∞.

Here ” →(d) ” means the convergence in the sense of finite dimensional distributions. We also use the notation, if f is a function on R,

fn= ∞ X j=−∞ f(j n)1(nj, j+1 n ], f + n,T = T X j=0 f(j n)1(nj, j+1 n ], f − n,T = −1 X j=−T f(j n)1(nj, j+1 n ] (20) and also f+ n = fn,∞+ , fn−= fn,∞− .

We have the following Non-Central Limit Theorem.

Theorem 1 Let f ∈ |H| such that fn± ∈ |H| for every n ≥ 1 and assume that |fn− f ||H|→n→∞ 0

and for every n

|fn,T± − fn±||H|→T→∞ 0. Then, as n → ∞, 1 nH X j∈Z f µj n ¶ g(ξj) →(d) ck Z R f(u)dZHk(u) (21)

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Proof: Let us prove first that the sum S(n) := 1 nH X j∈Z f µj n ¶ g(ξj)

is convergent in L2(Ω). We regard only the right tail part, the left part being similar. Using

the relation E[Hk1(ξi)Hk2(ξj)] = δk1,k2k1!r(i, j) k1 we get E£S(n)2¤ = 1 n2H X j1,j2∈Z f µj 1 n ¶ f µj 2 n ¶X∞ l=k c2lE[Hl(ξj1)Hl(ξj2)] = 1 n2H X j1,j2∈Z f µj 1 n ¶ f µj 2 n ¶X∞ l=k c2ll!r(j1− j2)l = 1 n2H X j16=j2 f µj 1 n ¶ f µj 2 n ¶X∞ l=k c2ll!|j1− j2| (2H−2)l k + 1 n2H X j∈Z f µj n ¶2 ∞X l=k l!c2l

Now, since for |j1− j2| ≥ 1 and for l ≥ k one has |j1− j2|

(2H−2)l

k ≤ |j1− j2|2H−2, and since the sum C0:=Pj∈Nc2ll! is convergent, we obtain

ES(n)2 ≤ C0 n2 X j16=j2 ¯ ¯ ¯ ¯f µ j1 n ¶¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯f µ j2 n ¶¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ j1 n − j2 n ¯ ¯ ¯ ¯ 2H−2 + C0 n2H X j∈Z f µ j n ¶2 ≤ C0 X j16=j2 ¯ ¯ ¯ ¯f µ j1 n ¶¯¯ ¯ ¯ ¯ ¯ ¯ ¯f µ j2 n ¶¯¯ ¯ ¯ Z j1+1 n j1 n Z j2+1 n j2 n |u − v|2H−2dudv+ C0 n2H X j∈Z f µ j n ¶2 = C0kfnk2|H|. (22)

In the same way we will obtain

E   ¯ ¯ ¯ ¯ ¯ ¯ 1 nH p2 X j=p1+1 f µj n ¶ g(ξj) ¯ ¯ ¯ ¯ ¯ ¯ 2  ≤ C0kfn,p+ 2− f + n,p1k 2 |H|

and this tend to zero as p1, p2 → ∞ by assumption.

Let us prove now that the sequence S(n) converges to the Wiener integralRRf(u)dZ k H(u)

when n tends to infinity. Here we follow the arguments [17]. Let us choose a sequence fr, r≥ 1

of elementary functions such that fr

r→∞f with respect to norm |H| (this is possible because

this space is complete) and denote by fnr =X j∈Z fr µj n ¶ 1(j n, j+1 n ] , r, n≥ 1.

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Note first that from a result in Bilingsley [5], it suffices to show that: Z R fr(u)dZk H(u) →(d) Z R f(u)dZk H(u), r → ∞, (23) Sr(n) := 1 nH ∞ X j=−∞ fr(j n)g(ξj) → (d) Z R fr(u)dZk

H(u), n→ ∞, for every r ≥ 1, (24)

and

lim

r limn E

h

|Sr(n) − S(n)|2i= 0. (25)

Concerning (23), we have by the construction of the Wiener integral and the choice of the sequence fr, E "¯ ¯ ¯ ¯ Z R fr(u)dZHk(u) − Z R f(u)dZHk(u) ¯ ¯ ¯ ¯ 2# = kfr− f k2H≤ kfr− f k2|H|→r→∞0.

The convergence (24) follows from the convergence of the sequence ZHk,n given by (19) to the Hermite process Zk

H because (since fr is an elementary function) Sr(n) constitutes a finite

linear combination of instants of the process ZHk,n and RRf

r(u)dZk

H(u) represents (the same)

finite linear combination of instants of Zk

H. For (25), proceeding as for the proof of (22), we

obtain

E|Sr(n) − S(n)|2 ≤ const. |fr

n− fn|2|H|

and thus by a dominated convergence argument (see [17], Proposition 3.1) we get lim r limn E|S r(n) − S(n)|2 ≤ const. lim r limn |f r n− fn|2|H|= const. limr |fr− f |2|H|= 0.

Example 1 By using relation (11) and the fact that kf k|H| ≤ const.kf k

LH1(R) one can see

that for every t > 0 the function h(x) = 1(−∞,t]eλx belongs to L1(R)TL2(R) and thus to |H|

and the same holds for hn given by (20). Using the continuity of the function h and comments

after Theorem 3.2 in [17], we can see that the hypothesis of Theorem 1 are satisfied in the case of the function h. As a consequence, at each t, the Hermite Ornstein-Uhlenbeck process can be approximated in law by the sequence in the left side of (21).

Let us briefly recall the spectral domain approach; by this, we mean the use of the space bH of deterministic integrands given by (12) which involves Fourier transforms. In [17] it has been proved that, if k = 1 and the assumptions of Theorem 1 are satisfied with bH instead of |H|, then the limit result (21) holds. It is not difficult to see that the fractional Orstein-Uhlenbeck process satisfies the hypothesis in Theorem 3.2 in [17]. It would be interesting to generalize it to the case of functions of Hermite rank k.

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References

[1] J.M.P. Albin (1998): A note on the Rosenblatt distributions. Statistics and Probability Letters, 40(1), pag. 83-91.

[2] J.M.P. Albin (1998): On extremal theory for self similar processes. Annals of Probability, 26(2), pag. 743-793.

[3] E. Alos and D. Nualart (2001): Stochastic integration with respect to the fractional Brownian motion. Stochastics and Stochastics Reports.

[4] E. Alos, O. Mazet and D. Nualart (2001): Stochastic calculus with respect to Gaussian processes. Annals of Probability, 29, pag. 766-801.

[5] P. Bilingsley (1968): Convergence of Probability Measures. Wiley, New York.

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