Limit theorems for integral functionals of Hermite-driven processes

HERMITE-DRIVEN PROCESSES

VALENTIN GARINO, IVAN NOURDIN, DAVID NUALART, AND MAJID SALAMAT

Abstract. Consider a moving average process X of the formX(t) = Rt

−∞ϕ(t−u)dZu,t≥0, where Z is a (non Gaussian) Hermite process of order q ≥2 and ϕ: R⁺ →R is sufficiently integrable. This paper investigates the fluctuations, as T → ∞, of integral functionals of the formt7→RT t

0 P(X(s))ds, in the case whereP is any given polynomial function. It extends a study initiated in [21], where only the quadratic caseP(x) =x² and the convergence in the sense of finite-dimensional distributions were considered.

1. Introduction

Hermite processes occur naturally when we consider limits of partial sums associated with long-range dependent stationary series. They have become increasingly popular in the recent literature, see for example the book [16]

by Pipiras and Taqqu, in particular section 4.11, which contains bibliograph- ical notes on their history and recent developments. They form a family of stochastic processes, indexed by an integerq≥1 and a self-similarity index H∈(¹₂,1), called the Hurst parameter, that contains the fractional Brown- ian motion (q = 1) and the Rosenblatt process (q = 2) as particular cases.

We refer the reader to Section 2.2 and the references therein for a precise definition of the Hermite processes. Of primary importance in the sequel is the parameterH0, given in terms ofH andq by

(1) H0= 1− 1−H

q ∈(1− 1 2q,1).

The goal of the present paper is to investigate the fluctuations, asT → ∞, of the family of stochastic processes

(2) t7→

Z T t 0

P(X(s))ds, t∈[0,1] (say),

in the case whereP(x) is a polynomial function andX is a moving average process of the form

(3) X(t) =

Z t

−∞

ϕ(t−u)dZ_u, t≥0,

2000Mathematics Subject Classification. 60F05, 60G22, 60H05, 60H07.

Key words and phrases. Hermite processes, chaotic decomposition, fractional Brownian motion (fBm), multiple Wiener-Itˆo integrals.

1

withZ a Hermite process andϕ:R+→Ra sufficiently integrable function.

We note that integral functionals such as (2) are often encountered in the context of statistical estimation, see e.g. [21] for a concrete example.

Let us first consider the case whereq= 1, that is to say the case where Z is the fractional Brownian motion. Note that this is the only case whereZ is Gaussian, making the studya priori much simpler and more affordable. By linearity and passage to the limit, the processX is also Gaussian. Moreover, it is stationary, since the quantityE[X(t)X(s)] =:ρ(t−s) only depends on t−s. For simplicity and without loss of generality, assume that ρ(0) = 1, that is,X(t) has variance 1 for anyt. As is well-known since the eighties (see [5, 8, 19]), the fluctuations of (2) heavily depends on the centered Hermite rank ofP, defined as the integerd≥1 such that P decomposes in the form

(4) P =E[P(X(0))] +

∞

X

k=d

akHk,

with Hk thekth Hermite polynomials and ad6= 0. (Note that the sum (4) is actually finite, sinceP is a polynomial, so that #{k: a_k6= 0}<∞.)

The first result of this paper concerns the fractional Brownian motion.

Even if it does not follow directly from the well-known results of Breuer- Major [5], Dobrushin-Major [8] and Taqqu [19], the limits obtained are somehow expected. In particular, the threshold H = 1− _2d¹ is well known to specialists. However, the proof of this result is not straightforward, and requires several estimations which are interesting in themselves.

Theorem 1. Let Z be a fractional Brownian motion of Hurst index H ∈ (¹₂,1), and let ϕ∈L¹(R+)∩L^H1(R+). Consider the moving average process Xdefined by (3) and assume without loss of generality thatVar(X(0)) = 1(if not, it suffices to multiplyϕby a constant). Finally, let P(x) =PN

n=0anxⁿ be a real-valued polynomial function, and let d ≥ 1 denotes its centered Hermite rank.

(1) If d≥2 and H∈(¹₂,1−_2d¹) then

(5) T⁻¹²

Z T t 0

P(X(s))−E[P(X(s))]

ds

t∈[0,1]

converges in distribution in C([0,1]) to a standard Brownian mo- tionW, up to some multiplicative constant C₁ which is explicit and depends only on ϕ, P and H.

(2) If H∈(1−_2d¹ ,1)then

(6) T^d(1−H)−1

Z T t 0

P(X(s))−E[P(X(s))]

ds

t∈[0,1]

converges in distribution inC([0,1]) to a Hermite process of indexd and Hurst parameter1−d(1−H), up to some multiplicative constant C₂ which is explicit and depends only on ϕ, P and H.

Now, let us consider the non-Gaussian case, that is, the case whereq≥2.

As we will see, the situation is completely different, both in the results obtained (rather unexpected) and in the methods used (very different from the Gaussian case). Let L > 0. We define S_L to be the set of bounded functions l : R+ → R such that y^Ll(y) → 0 as y → ∞. We observe that S_L ⊂ L¹(R+)∩L^H1(R+) for any L > 1. We can now state the following result.

Theorem 2. LetZ be a Hermite process of orderq≥2and Hurst parameter H∈(¹₂,1), and letϕ∈ S_Lfor some L >1. RecallH0 from (1) and consider the moving average processXdefined by (3). Finally, letP(x) =PN

n=0a_nxⁿ be a real-valued polynomial function. Then, one and only one of the following two situations takes place atT → ∞:

(i) If q is odd and if a_n6= 0 for at least one odd n∈ {1, . . . , N}, then T^−H0

Z T t 0

P(X(s))−E[P(X(s))]

ds

t∈[0,1]

converges in distribution inC([0,1])to a fractional Brownian motion of parameterH1 :=H0, up to some multiplicative constantK1 which is explicit and depends only onϕ, P, q and H, see Remark 4.

(ii) If q is even, or if q is odd and an = 0 for all odd n ∈ {1, . . . , N}, then

T^1−2H0 Z T t

0

P(X(s))−E[P(X(s))]

ds

t∈[0,1]

converges in distribution inC([0,1])to a Rosenblatt process of Hurst parameter H2 := 2H0 −1, up to some multiplicative constant K2

which is explicit and depends only onϕ, P, q and H, see Remark 4.

Remark 3. Whether in Theorem 1 or Theorem 2, the multiplicative con- stants appearing in the limit can be all given explicitly by following the respective proofs. For example, the constant K₁ and K₂ of Theorem 2 are given by the following intricate expressions:

K1 =

N

X

n=3,nodd

ancⁿ_H,qKϕ,n,1

K₂ =

N

X

n=2

a_ncⁿ_H,qK_ϕ,n,2+a₁I{q=2}

Z

R+

ϕ(ν)dν with

K_ϕ,n,i= X

α∈An,q,nq−2|α|=i

CαKϕ,α,H0

c_Hi,i , i= 1,2,

where the sets and constants in the previous formula are defined in Sections 2, 3 and 4.

Remark 4. Note that, unlike the case of a fractional Brownian motion X, where the limit depends on the Hermite rank of the polynomial P, here the Hermite rank of P plays no role and the limit depends on the parity of the non-vanishing coefficients of P. This is not really surprising in our non-Gaussian context, since the Hermite rank of P is defined by means of its decomposition into Hermite polynomials, and these latter polynomials only have good probabilistic properties when evaluated inGaussianrandom variables.

We note that our Theorem 2 contains as a very particular case the main result of [21], which corresponds to the choiceP(x) =x² and thus situation (ii). Moreover, let us emphasize that our Theorem 2 not only studies the convergence of finite-dimensional distributions as in [21], but also provides a functionalresult.

Because the employed method is new, let us sketch the main steps of the proof of Theorem 2, by using the classical notation of the Malliavin calculus (see Section 2 for any unexplained definition or result); in particular we writeI_p^B(h) to indicate thepth multiple Wiener-Itˆo integral of kernelhwith respect to the standard (two-sided) Brownian motionB.

(Step 1) In Section 3, we represent the moving average processX as a qth multiple Wiener-Itˆo integral with respect to B:

X(t) =cH,qI_q^B(g(t,·)),

wherecH,q is an explicit constant and the kernelg(t,·) is given by (7) g(t, ξ₁, . . . , ξ_q) =

Z _t

−∞

ϕ(t−v)

q

Y

j=1

(v−ξ_j)^H0−

3

+ 2dv,

for ξ₁, . . . , ξ_q ∈ R, t ≥ 0. Thanks to this representation, we compute in Lemma 5 the chaotic expansion of thenth power of X(t) for anyn≥2 and t >0, and obtain an expression of the form

Xⁿ(t) =cⁿ_H,q X

α∈A_n,q

C_αI_nq−2|α|^B (⊗_α(g(t,·), . . . , g(t,·))),

where we have used the novel notation⊗_α(g(t,·), . . . , g(t,·)) to indicate iter- ated contractions whose precise definition is given in Section 2.1, and where C_α are combinatorial constants and the sum runs over a familyA_n,q of suit- able multi-indices α = (αij,1≤i < j ≤n). As an immediate consequence, we deduce that our quantity of interest can be decomposed as follows:

Z T t 0

P(X(s))−E[P(X(s))]

ds=a₀ Z T t

0

X(s)ds (8)

+

N

X

n=2

ancⁿ_H,q X

α∈An,q,nq−2|α|≥1

Cα

Z T t 0

I_nq−2|α|^B (⊗_α(g(s,·), . . . , g(s,·)))ds.

(Step 2) In Proposition 6, we compute an explicit expression for the iterated contractions⊗_α(g(t,·), . . . , g(t,·)) appearing in the right-hand side of (8), by using that gis given by (7).

To ease the description of the remaining steps, let us now set (9) Fn,q,α,T(t) =

Z T t 0

I_nq−2|α|^B (⊗_α(g(s,·), . . . , g(s,·)))ds.

(Step 3)AsT → ∞, we show in Proposition 7 that, ifnq−2|α|< _1−H^q , then T^{−1+(1−H0)(nq−2|α|)}Fn,q,α,T(t) converges in distribution to a Hermite process (whose order and Hurst index are specified) up to some multiplica- tive constant. Similarly, we prove in Proposition 9 that, if nq−2|α| ≥ 3, thenT^α0Fn,q,α,T(t) is tight and converges inL²(Ω) to zero, whereα0is given in (28).

(Step 4)By putting together the results obtained in the previous steps, the two convergences stated in Theorem 2 follow immediately.

To illustrate a possible use of our results, we study in Section 6 an exten- sion of the classical fractional Ornstein-Uhlenbeck process (see, e.g., Cherid- itoet al[7]) to the case where the driving process is more generally a Hermite process. To the best of our knowledge, there is very little literature devoted to this mathematical object, only [11, 17].

The rest of the paper is organized as follows. Section 2 presents some basic results about multiple Wiener-Itˆo integrals and Hermite processes, as well as some other facts that are used throughout the paper. Section 3 contains preliminary results. The proof of Theorem 1 (resp. Theorem 2) is given in Section 5 (resp. Section 4). In Section 6, we provide a complete asymptotic study of the Hermite-Ornstein-Uhlenbeck process, by means of Theorems 1 and 2 and of an extension of Birkhoff’s ergodic Theorem. Finally, Section 7 contains two technical results: a power counting theorem and a version of the Hardy-Littelwood inequality, which both play an important role in the proof of our main theorems.

2. Preliminaries on multiple Wiener-Itˆo integrals and Hermite processes

2.1. Multiple Wiener-Itˆo integrals and a product formula. A func- tion f :R^p → R is said to be symmetric if the following relation holds for all permutationσ ∈S(p):

f(t₁, . . . , t_p) =f(t_σ(1), . . . , t_σ(p)), t₁, . . . , t_p ∈R.

The subset ofL²(R^p) composed of symmetric functions is denoted byL²_s(R^p).

Let B = {B(t)}_t∈R be a two-sided Brownian motion. For any given f ∈L²_s(R^p) we consider the multiple Wiener-Itˆo integral of f with respect

toB, denoted by

I_p^B(f) = Z

R^p

f(t₁, . . . , t_p)dB(t₁)· · ·dB(t_p).

This stochastic integral satisfies E[I_p^B(f)] = 0 and E[I_p^B(f)I_q^B(g)] =1{p=q}p!hf, gi_L2(R^p)

forf ∈L²_s(R^p) and g ∈L²_s(R^q), see [10] and [13] for precise definitions and further details.

It will be convenient in this paper to deal with multiple Wiener-Itˆo inte- grals of possibly nonsymmetric functions. If f ∈ L²(R^p), we put I_p^B(f) = I_p^B( ˜f), where ˜f denotes the symmetrization off, that is,

f˜(x₁, . . . , x_p) = 1 p!

X

σ∈S(p)

f(x_σ(1), . . . , x_σ(p)).

We will need the expansion as a sum of multiple Wiener-Itˆo integrals for a product of the form

n

Y

k=1

I_q^B(h_k),

whereq≥2 is fixed and the functionsh_k belong to L²_s(R^q) fork= 1, . . . , n.

In order to present this extension of the product formula and to define the relevant contractions between the functions hi and hj that will naturally appear, we introduce some further notation. Let A_n,q be the set of multi- indices α= (αij,1≤i < j ≤n) such that, for eachk= 1, . . . , n,

X

1≤i<j≤n

αij1k∈{i,j}≤q.

Set|α|=P

1≤i<j≤nαij, β⁰_k=q− X

1≤i<j≤n

α_ij1_k∈{i,j}, 1≤k≤n and

(10) m:=m(α) =

n

X

k=1

β_k⁰=nq−2|α|.

For each 1≤i < j≤n, the integerαij will represent the number of variables inh_iwhich are contracted withh_jwhereas, for eachk= 1, . . . , n, the integer β_k⁰ is the number of variables in h_k which are not contracted. We will also write β_k=Pk

j=1β_j⁰ fork= 1, . . . , n and β₀ = 0. Finally, we set

(11) Cα= q!ⁿ

Qn

k=1β_k⁰!Q

1≤i<j≤nα_ij!.

With these preliminaries, for any element α ∈ A_n,q we can define the contraction ⊗_α(h₁, . . . , h_n) as the function of nq−2|α| variables obtained

by contracting αij variables between hi and hj for each couple of indices 1≤i < j ≤n. Define the collection (u^i,j)1≤i,j≤n,i6=j in the following way:

u^i,j =αmin(i,j),max(i,j). We then have

⊗_α(h₁, . . . , h_n)(ξ₁, . . . , ξ_nq−2|α|)

= Z

R^|α|

n

Y

k=1

h_k(s^k,1₁ , . . . , s^k,1_uk,1, . . . , s₁^k,n, . . . , s^k,n_uk,n, ξ_1+βk−1, . . . , ξ_βk) (12)

× Y

1≤i<j≤n

ds^i,j₁ . . . ds^i,j_ui,j

Whenn= 2, α has only one componentα_1,2 and ⊗_α(h₁, h₂) =h₁⊗_α1,2h₂ is the usual contraction of α1,2 indices betweenh1 and h2. Notice that the function⊗_α(h₁, . . . , h_n) is not necessarily symmetric.

Then, we have the following result.

Lemma 5. Let n, q ≥ 2 be some integers and let hi ∈ L²_s(R^q) for i = 1, . . . , n. We have

n

Y

k=1

I_q^B(h_k) = X

α∈An,q

C_αI_nq−2|α|^B (⊗_α(h₁, . . . , h_n)).

(13)

Proof. The product formula for multiple stochastic integrals (see, for in- stance, [14, Theorem 6.1.1], or formula (2.1) in [3] for n= 2) says that (14)

n

Y

k=1

I_q^B(h_k) =X

P,ψ

I_β^B0 1+···+β⁰_n

(⊗ⁿ_k=1h_k)_P,ψ ,

whereP denotes the set of all partitions {1, . . . , q}=Ji∪(∪k=1,...,n,k6=iIik), where for any i, j = 1, . . . , n, I_ij and I_ji have the same cardinalityα_ij,ψ_ij is a bijection between Iij and Iji and β_k⁰ = |J_k|. Moreover, (⊗ⁿ_k=1h_k)_P,ψ denotes the contraction of the indexes ` and ψij(`) for any`∈Iij and any i, j = 1. . . , n. Then, formula (13) follows form (14), by just counting the number of partitions, which is

n

Y

k=1

q!

Q

iorj6=kα_ij!β_k⁰!

and multiplying by the number of bijections, which isQ

1≤i<j≤nαij!.

Notice that whenn= 2, formula (13) reduces to the well-known formula for the product of two multiple integrals. That is, for any two symmetric functions f ∈L²_s(R^p) and g∈L²_s(R^q) we have

I_p^B(f)I_q^B(g) =

min(p,q)

X

r=0

r!

p r

q r

I_p+q−2r^B (f⊗_rg).

where, for 0≤r ≤min(p, q), f⊗_rg∈L²(R^p+q−2r) denotes the contraction of r coordinates betweenf and g.

2.2. Hermite processes. Fixq ≥1 and H ∈(¹₂,1). The Hermite process of indexqand Hurst parameterHcan be represented by means of a multiple Wiener-Itˆo integral with respect to B as follows, see e.g. [9]:

(15) Z^H,q(t) =c_H,q Z

R^q

Z

[0,t]

q

Y

j=1

(s−x_j)^H0−

3 2

+ dsdB(x₁)· · ·dB(x_q), t∈R. Here,x+= max{x,0}, the constantcH,q is chosen to ensure that

Var(Z^H,q(1)) = 1, and

H0 = 1−1−H

q ∈ 1− 1 2q,1

.

Note that Z^H,q is self-similar of index H. When q = 1, the process Z^H,1 is Gaussian and is nothing but the fractional Brownian motion with Hurst parameter H. For q≥2, the processes Z^H,q are no longer Gaussian:

they belong to the qth Wiener chaos. The process Z^H,2 is known as the Rosenblatt process.

Let|H| be the following class of functions:

|H|=n

f :R→R

Z

R

Z

R

|f(u)||f(v)||u−v|^2H−2dudv <∞o . Maejima and Tudor [9] proved that the stochastic integralR

Rf(u)dZ^H,q(u) with respect to the Hermite process Z^H,q is well defined whenf belongs to

|H|. Moreover, for any orderq ≥1, indexH ∈(¹₂,1) and function f ∈ |H|, Z

R

f(u)dZ^H,q(u) (16)

= c_H,q Z

R^q



 Z

R

f(u)

q

Y

j=1

(u−ξ_j)^H0−

3 2

+ du



dB(ξ₁)· · ·dB(ξ_q).

As a consequence of the Hardy-Littlewood-Sobolev inequality featured in [1], we observe thatL¹(R)∩L^H1(R)⊂ |H|.

3. Chaotic decomposition of R_{T t}

0 P(X(s))ds

Assumeϕ∈ |H|and q ≥1. Using (16) and bearing in mind the notation and results from Section 2, it is immediate thatX can be written as (17) X(t) =c_H,qI_q^B(g(t, .)),

whereg(t, .) is given by (18) g(t, ξ₁, . . . , ξ_q) =

Z _t

−∞

ϕ(t−v)

q

Y

j=1

(v−ξ_j)^H₊^0−3/2dv, and c_H,q is defined as in (15).

3.1. Computing the chaotic expansion of X(t)ⁿ when n≥2. Let us denote byA⁰_n,q the set of elementsα∈A_n,q such thatnq−2|α|= 0 andA¹_n,q will be the set of elements α ∈ A_n,q such that nq−2|α| ≥ 1. Notice that when nq is odd,A⁰_n,q is empty. Using (13), we obtain the following formula for the expectation of thenth power (n≥2) ofX given by (3):

(19) E[X(t)ⁿ] = (c_H,q)ⁿ X

α∈A⁰_n,q

C_αI_nq−2|α|^B (⊗_α(g(t,·), . . . , g(t,·))).

We observe in particular thatE[X(t)ⁿ] = 0 whenevernq is odd. From (13) and (19), we deduce for n≥2 that

(20) X(t)ⁿ−E[X(t)ⁿ] = (c_H,q)ⁿ X

α∈A¹_n,q

C_αI_nq−2|α|^B (⊗_α(g(t,·), . . . , g(t,·))).

To clarify this formula, let us write down detailed a expression in the casesn= 2 and n= 3. Whenn= 2, the right-hand side of (20) is

(cH,q)²

q−1

X

r=0

r!

q r

2

I_2q−2r^B (g(t,·)⊗_rg(t,·)),

because α has just one componentα1,2 =:r and condition α∈A¹_n,q means 0≤r≤q−1. Forn= 3, we have

A_3,q ={(α_1,2, α_1,3, α_2,3) :α_1,2+α_1,3≤q, α_1,2+α_2,3≤q, α_1,3+α_2,3 ≤q}

and the right-hand side of (20) is (cH,q)³ X

α∈A3,q:3q−2|α|≥1

CαI_3q−2|α|(⊗_α(g(t,·), g(t,·), g(t,·)), where

C_α = (q!)³

α_1,2!α_1,3!α_2,2!(q−α_1,2−α_1,3)!(q−α_1,2−α_2,3)!(q−α_1,3−α_1,3)!. In this case, the contraction⊗_α(g(t,·), g(t,·), g(t,·)) is the function of 3q− 2|α|variables defined by

Z

R^|α|

g(•, s, u)g(?, s, v))g(◦, u, v)dsdudv,

withs= (s1, . . . , sα1,2),u= (u1, . . . , uα1,3) and v= (v1, . . . , vα2,3).

From (20) we obtain

Z T t 0

P(X(s))−E[P(X(s))]

ds=a1

Z T t 0

X(s)ds (21)

+

N

X

n=2

a_n(c_H,q)ⁿ X

α∈A¹_n,q

C_α Z T t

0

I_nq−2|α|^B (⊗_α(g(s,·), . . . , g(s,·)))ds.

3.2. Expressing the iterated contractions of g. We now compute an explicit expression for the iterated contractions appearing in (21).

Proposition 6. Fix n≥2, q≥1 and α∈A_n,q. We have

⊗_α(g(t, .), . . . , g(t, .))(ξ) =β(H₀− 1

2,2−2H₀)^|α|

× Z

(−∞,t]ⁿ

dv₁. . . dv_n

n

Y

k=1

ϕ(t−v_k) Y

1≤i<j≤n

|v_i−v_j|^{(2H0−2)αij}

×

n

Y

k=1 βk

Y

`=1+βk−1

(vk−ξ`)^H0−

3 2

+ ,

with the convention β0 = 0.

Proof. The proof is a straightforward consequence of the following identity (22)

Z

R

(v−ξ)^H₊^0−3/2(w−ξ)₊^H0−3/2dξ=β(H₀−1

2,2−2H₀)|v−w|^(2H0−2),

whose proof is elementary, see e.g. [4].

4. Proof of Theorem 2

We are now ready to prove Theorem 2. To do so, we will mostly rely on the forthcoming Proposition 7, which might be a result of independent interest by itself, and which studies the asymptotic behavior ofF_n,q,α,T^B given by (9).

We will denote by f.d.d. the convergence in law of the finite-dimensional distributions of a given process. Notice that the hypothesis on ϕ is a bit weaker than the one in the main theorem, the fact that ϕ ∈ S_L being required in the forthcoming Proposition 9.

Proposition 7. Fix n ≥ 2, q ≥ 1 and α ∈ An,q. Assume the function ϕ belongs to L¹(R+)∩L^H1(R+), recall H₀ from (1) and let m be defined as in (10). Finally, assume that 2m < _1−H^q (which is automatically satisfied when m= 1 or m= 2). Then, asT → ∞,

(23) (T^{−1+(1−H0)m}F_n,q,α,T(t))_t∈[0,1]^f.d.d.−→

CαKϕ,α,H0

c_H(m),m Z^H(m),m(t)

t∈[0,1]

,

where Z^H(m),m denotes the mth Hermite process of Hurst index H(m) = 1− ^m_q(1−H) and the constants C_α and K_ϕ,α,H0 are defined in (11) and (24), respectively. Furthermore, {T^{−1+(1−H0)m}(Fn,q,α,T(t))t∈[0,1], T > 0} is tight in C([0,1]).

Remark 8. Note that for m₁ < m₂ the chaos of order m₁ dominates the chaos of order m₂.

Proof of Proposition 7. Letn≥2,q ≥1 andα∈An,q.

Step 1: We will first show the convergence (23). We will make several change of variables in order to transform the expression ofFn,q,α,T(t). By means of an application of stochastic’s Fubini’s theorem, we can write

Fn,q,α,T(t) =Cα

Z

R^m

ΨT(ξ1, . . . , ξm)dB(ξ1)· · ·dB(ξm), where

Ψ_T(ξ₁, . . . , ξ_m) :=T^{−1+m(1−H0)} Z _{T t}

0

ds Z

(−∞,s]ⁿ

dv₁· · ·dv_n

n

Y

k=1

ϕ(s−v_k)

× Y

1≤i<j≤n

|v_i−v_j|^{(2H0−2)αij}

n

Y

k=1 βk

Y

`=1+βk−1

(v_k−ξ_`)^H0−

3 2

+ .

Using the change of variables s → T s and vk → T s−vk, 1 ≤ k ≤ n, we obtain

Ψ_T(ξ₁, . . . , ξ_m) :=T^m(1−H0) Z t

0

ds Z

(−∞,T s]ⁿ

dv₁· · ·dv_n

n

Y

k=1

ϕ(T s−v_k)

× Y

1≤i<j≤n

|v_i−vj|^{(2H0−2)αij}

n

Y

k=1 βk

Y

`=1+βk−1

(vk−ξ`)^H0−

3 2

+

=T^−m2 Z t

0

ds Z

[0,∞)ⁿ

dv₁· · ·dv_n

n

Y

k=1

ϕ(v_k)

× Y

1≤i<j≤n

|v_i−v_j|^{(2H0−2)αij}

n

Y

k=1 βk

Y

`=1+βk−1

(s−v_k T − ξ_`

T)^H0−

3 2

+ .

By the scaling property of the Brownian motion, the processes (Fn,q,α,T(t))α∈A_n,q,2≤n≤N,t∈[0,1]

and

(Fbn,q,α,T(t))α∈A_n,q,2≤n≤N,t∈[0,1]

have the same probability distribution, where Fbn,q,α,T(t) =Cα

Z

R^m

ΨbT(ξ1, . . . , ξm)dB(ξ1)· · ·dB(ξm)

Ψb_T(ξ₁, . . . , ξ_m) :=

Z t 0

ds Z

(−∞,0]ⁿ

dv₁· · ·dv_n

n

Y

k=1

ϕ(v_k)

× Y

1≤i<j≤n

|v_i−v_j|^{(2H0−2)αij}

n

Y

k=1 βk

Y

`=1+βk−1

(s−v_k

T −ξ_`)^H0−

3 2

+ .

Set

Ψ(ξb 1, . . . , ξm) :=Kϕ,α,H0

Z t 0

ds

m

Y

`=1

(s−ξ`)^H0−

3 2

+ ,

where

(24) K_ϕ,α,H0 = Z

Rⁿ+

dv1· · ·dvn n

Y

k=1

ϕ(v_k) Y

1≤i<j≤n

|v_i−vj|^{(2H0−2)αij}. Notice that, by Lemma 15, Kϕ,α,H0 is well defined. We claim that

(25) lim

T→∞Ψb_T =Ψ,b

where the convergence holds inL²(R^m). This will imply the convergence in L²(Ω) of Fb_n,q,α,T(t), asT → ∞ to a Hermite process of orderm, multiplied by the constant CαKϕ,α,H0.

Proof of (25): It suffices to show that the inner products hΨb_T,Ψb_Ti_L2(R^m)

and hΨb_T,Ψib _L2(R^m) converge, as T → ∞, to kΨkb ²_L2(R^m)=K_ϕ,α,H² ₀β(H0−1

2,2−2H0)^m Z

[0,t]²

dsds⁰|s−s⁰|^(2H0−2)m, which is finite because m < _2(1−H¹

0) = _2(1−H)^q . We will show the conver- gence of hΨbT,ΨbTi_L2(R^m) and the second term can be handled by the same arguments. We have

kΨb_Tk²_L2(R^m)= Z

[0,t]²

dsds⁰ Z

R²ⁿ+

dv₁· · ·dv_ndv₁⁰ · · ·dv_n⁰

×

n

Y

k=1

ϕ(vk)ϕ(v⁰_k) Y

1≤i<j≤n

|v_i−vj|^{(2H0−2)αij}|v_i⁰−v_j⁰|^{(2H0−2)αij}

×

n

Y

k=1

β(H0− 1

2,2−2H0)^βk|s−s⁰−vk−v_k⁰

T |^(2H0−2)βk. Let us first show that given wk∈R, 1≤k≤n,

(26) lim

T→∞

Z

[0,t]²

dsds⁰

n

Y

k=1

|s−s⁰−wk

T |^(2H0−2)βk = Z

[0,t]²

dsds⁰|s−s⁰|^(2H0−2)m and, moreover,

(27) sup

wk∈R,1≤k≤n

Z

[0,t]²

dsds⁰

n

Y

k=1

|s−s⁰−w_k|^(2H0−2)βk <∞.

By the dominated convergence theorem and using Lemma 15, (26) and (27) imply (25).

To show (26), choose such that |w_k|/T < , 1 ≤ k ≤ n, for T large enough (depending on the fixed w_k’s). Then, we can write

Z

[0,t]²

dsds⁰

n

Y

k=1

|s−s⁰−wk

T |^(2H0−2)βk − |s−s⁰|^(2H0−2)m

≤t Z

|ξ|>2

dξ

n

Y

k=1

|ξ−w_k

T |^(2H0−2)βk − |ξ|^(2H0−2)m + 2t sup

|w_k|<

Z

|ξ|≤2

dξ

n

Y

k=1

|ξ−wk|^(2H0−2)βk :=B₁+B₂.

The term B₁ tends to zero a T → ∞, for each > 0. On the other hand, the termB2 tends to zero as →0. Indeed,

B2 = 2t^(2H0−2)m+1 sup

|w_k|<1

Z

|ξ|≤2

dξ

n

Y

k=1

|ξ−w_k|^(2H0−2)βk.

Note that the above supremum is finite because the function (w1, . . . , w_k)→ R

|ξ|≤2dξQn

k=1|ξ−w_k|^(2H0−2)βk is continuous.

Property (27) follows immediately from the fact that the function (w₁, . . . , w_k)→

Z

[0,t]²

dsds⁰

n

Y

k=1

|s−s⁰−w_k|^(2H0−2)βk is continuous and vanishes as |(w₁, . . . , w_k)|tends to infinity.

We have H0 = 1− ^1−H_q = 1− ^1−H_m^(m) with H(m) as above. As a re- sult, we obtain the convergence of the finite-dimensional distributions of T^{−1+(1−H0)m}F_n,q,α,T(t) to those the mth Hermite process Z^H(m),m multi- plied by the constant ^C_c^αKϕ,α,H0

H(m),m .

Step 2: Tightness. Fix 0 ≤ s < t ≤ 1. To check that tightness holds in C([0,1]), let us compute the squared L²(Ω)-norm

Φ_T :=T^{−1+(1−H0)m}E(|F_n,q,α,T(t)−F_n,q,α,T(s)|²).

Proceeding as in the first step of the proof, we obtain Ψ_T =E

Z

R^m

dB(ξ₁)· · ·dB(ξ_m) Z t

s

du Z

Rⁿ+

dv₁· · ·dv_n

n

Y

k=1

ϕ(v_k)

× Y

1≤i<j≤n

|v_i−vj|^{(2H0−2)αij}

n

Y

k=1 β_k

Y

`=1+βk−1

(u−vk

T −ξ`)^H0−

3 2

+

2!

≤m!

Z

R^m

dξ1· · ·dξm

Z t s

du Z

Rⁿ+

dv1· · ·dvn n

Y

k=1

ϕ(vk)

× Y

1≤i<j≤n

|v_i−v_j|^{(2H0−2)αij}

n

Y

k=1 βk

Y

`=1+βk−1

(u−v_k

T −ξ_`)^H0−

3 2

+

2

.

Using (22) yields Ψ_T ≤m!

Z

[s,t]²

dudu⁰ Z

R²ⁿ+

dv1· · ·dvndv₁⁰ · · ·dv_n⁰

×

n

Y

k=1

ϕ(v_k)ϕ(v_k⁰) Y

1≤i<j≤n

|v_i−v_j|^{(2H0−2)αij}|v_i⁰−v_j⁰|^{(2H0−2)αij}

×

n

Y

k=1

β(H0−1

2,2−2H0)^βk|u−u⁰−v_k−v_k⁰

T |^(2H0−2)βk

≤m!(t−s) Z 1

−1

dξ Z

R²ⁿ+

dv1· · ·dvndv⁰₁· · ·dv⁰_n

×

n

Y

k=1

ϕ(v_k)ϕ(v_k⁰) Y

1≤i<j≤n

|v_i−v_j|^{(2H0−2)αij}|v_i⁰−v_j⁰|^{(2H0−2)αij}

×

n

Y

k=1

β(H0−1

2,2−2H0)^βk|ξ−v_k−v_k⁰

T |^(2H0−2)βk

≤C(t−s).

Then the equivalence of all L^p(Ω)-norms, p ≥ 2, on a fixed Wiener chaos, also known as the hypercontractivity property, allows us to conclude the

proof of the tightness.

We will make use of the notation

(28) α₀ = (1−2H₀)1_{nq is even^}−H₀1_{nq is odd}.

Proposition 9. Fix n, q ≥ 2 and α ∈ A_n,q, assume that the function ϕ belongs to S_L for some L > 1 and that m ≥ 3. Then for any t ∈ [0,1], T^α0F_n,q,α,T(t) converge inL²(Ω)to zero asT → ∞; furthermore, the family {(F_n,q,α,T(t))t∈[0,1], T >0} is tight in C([0,1]).

Proof. If (2H₀−2)m >−1, by Proposition 7, we know thatT^{−1+m(1−H0)}F_n,q,α,T(t) converges to zero inL²(Ω) asT → ∞. This implies the convergence to zero inL²(Ω) as T → ∞of T^−α0Fn,q,α,T(t) because −1 +m(1−H0) > α0. We should then concentrate on the case (2H₀−2)m≤ −1. Once again, we shall divide the proof in two steps:

Step 1: Let us first prove the convergence in L²(Ω). Fix α∈A_n,q. We are going to show that

Tlim→∞T^2α0E |F_n,q,α,T(t)|²

= 0.

We know that

T^2α0E |F_n,q,α,T(t)|²

=T^2α0m!× Z

[0,T t]

ds⊗_α(g(s,·), . . . , g(s,·))

2

L²(R^m)

. In view of the expression for the contractions obtained in Proposition 7, it suffices to show that

Tlim→∞T^2α0 Z

[0,T t]²

dsds⁰ Z

R^m

Z

(−∞,s]ⁿ

Z

(−∞,s⁰]ⁿ

dv1· · ·dvndv₁⁰ · · ·dv_n⁰dξ1· · ·dξm

×

n

Y

k=1

ϕ(s−v_k)ϕ(s⁰−v_k⁰) Y

1≤i<j≤n

|v_i−vj|^{(2H0−2)αij}|v⁰_i−v⁰_j|^{(2H0−2)αij}

×

n

Y

k=1 βj

Y

`=1+βj−1

(vk−ξ`)^H0−

3 2

+

βj

Y

`=1+βj−1

(v⁰_k−ξ`)^H0−

3 2

+ = 0.

Integrating in the variables ξ’s and using (22), it remains to show that

Tlim→∞T^2α0 Z

[0,T t]²

dsds⁰ Z

(−∞,s]ⁿ

Z

(−∞,s⁰]ⁿ

dv1· · ·dvndv⁰₁· · ·dv⁰_n

×

n

Y

k=1

ϕ(s−v_k)ϕ(s⁰−v⁰_k) Y

1≤i<j≤n

|v_i−v_j|^{(2H0−2)αij}|v⁰_i−v⁰_j|^{(2H0−2)αij}

×

n

Y

k=1

|v_k−v_k⁰|^(2H0−2)βk = 0.

Set

ΦT :=T^2α0 Z

[0,T t]²

dsds⁰ Z

(−∞,s]ⁿ

Z

(−∞,s⁰]ⁿ

dv1· · ·dvndv₁⁰ · · ·dv_n⁰

×

n

Y

k=1

ϕ(s−v_k)ϕ(s⁰−v_k⁰) Y

1≤i<j≤n

|v_i−v_j|^{(2H0−2)αij}|v⁰_i−v⁰_j|^{(2H0−2)αij}

×

n

Y

k=1

|v_k−v_k⁰|^(2H0−2)βk.

Making the change of variableswk =s−vk, w_k⁰ =s⁰−v_k⁰ fork= 1, . . . , n, yields

Φ_T =T^2α0 Z

[0,T t]²

dsds⁰ Z

Rⁿ+

Z

[0,∞)ⁿ

dw₁· · ·dw_ndw₁⁰ · · ·dw_n⁰

×

n

Y

k=1

ϕ(wk)ϕ(w⁰_k) Y

1≤i<j≤n

|w_i−wj|^{(2H0−2)αij}|w⁰_i−w_j⁰|^{(2H0−2)αij}

×

n

Y

k=1

|s−s⁰−w_k+w_k⁰|^(2H0−2)βk.

Now we use Fubini’s theorem and make the change of variables s−s⁰ =ξ to obtain

ΦT =tT^2α0+1 Z

R²ⁿ+

dw1· · ·dwndw⁰₁· · ·dw⁰_n

×

n

Y

k=1

|ϕ(w_k)ϕ(w⁰_k)| Y

1≤i<j≤n

|w_i−w_j|^{(2H0−2)αij}|w_i⁰−w⁰_j|^{(2H0−2)αij}

× Z tT

−tT

dξ

n

Y

k=1

|ξ−w_k+w⁰_k|^(2H0−2)βk. We shall distinguish again two subcases:

Case (2H0−2)m <−1: Notice that the exponent 2α₀+ 1 is negative:

(i) Ifnq is even, then α0 = 1−2H0 and 2α0+ 1 = 3−4H0 <0 becauseH₀> ³₄.

(ii) If nq is odd, then α0 =−H₀ and

2α0+ 1 = 1−2H0 <0.

Therefore, in order to show that lim_T→∞Φ_T = 0, it suffices to check that J :=

Z

R²ⁿ

dw1· · ·dwndw₁⁰ · · ·dw_n⁰

n

Y

k=1

|ϕ(w_k)ϕ(w⁰_k)|

× Y

1≤i<j≤n

|w_i−wj|^{(2H0−2)αij}|w⁰_i−w⁰_j|^{(2H0−2)αij}

× Z

R

dξ

n

Y

k=1

|ξ−w_k+w⁰_k|^(2H0−2)βk <∞, (29)

where, by conventionϕ(w) = 0 ifw <0. We will apply the Power Counting Theorem 14 to prove that this integral is finite. We consider functions on R²ⁿ⁺¹ with variables{(w_k)k≤n,(w_k⁰)k≤n, ξ}. The set of linear functions is

T ={ω_k, ω⁰_k,1≤k≤n} ∪ {w_i−w_j, w⁰_i−w⁰_j,1≤i < j≤n}

∪ {ξ−w_k+w⁰_k,1≤k≤n}.

The corresponding exponents (µ_M, ν_M) for each M ∈T are (0,−L) for the linear functions w_k and w⁰_k(taking into account that ϕ∈ S_L), (2H0−2)αij

for each function of the formw_i−w_j orw⁰_i−w⁰_j and (2H₀−2)β_k for each function of the form ξ−w_k+w⁰_k.

Then J <∞, provided conditions (a) and (b) are satisfied.

•Verification of (b): Let W ⊂T be a linearly independent proper subset of T, and

d∞= 2n+ 1−dim(Span(W)) + X

M∈T\(Span(W)∩T)

ν_M.