Weak convergence to the fractional Brownian sheet in Besov spaces

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Weak convergence to the fractional Brownian sheet in Besov spaces

Ciprian A. Tudor

Abstract. In this paper we study the problem of the approximation in law of the fractional Brownian sheet in the topology of the anisotropic Besov spaces. We prove the convergence in law of two families of processes to the fractional Brownian sheet:

the first family is constructed from a Poisson procces in the plane and the second family is defined by the partial sums of two sequences of real independent fractional brownian motions.

Keywords: Fractional Brownian motion, weak convergence, Besov spaces.

Mathematical subject classification: 60F05, 60G15.

1 Introduction

LetT = [0,1] the unit interval and(W_t^α)t∈T a fractional Brownian motion of Hurst parameterα ∈ (0,1)on some probability space(,F, P ). That is,W^α is a centered Gaussian process, starting from zero and its covariance is given by

R(t, s)=E B_t^αB_s^α

= 1 2

t²^α+s²^α− |t−s|²^α .

Recall that the fractional Brownian motion of Hurst parameter α ∈ (0,1) admits a Wiener integral representation with respect toW of the form

W_t^α = _t

0

Kα(t, s)dWs (1)

whereKα is the kernel defined on the set{0< s < t}and it is given by Kα(t, s)=dα(t−s)^α−¹² +dα(1

2 −α) _t

s

(u−s)^α−³²

1−s u

¹₂_−α du,

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Received 14 March 2003.

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withdα the following normalizing constant

dα =

2α(³₂−α) (α+ ¹₂)(2−2α)

1 2

.

Let now(Wu,v)_(u,v)∈T² a Brownian sheet. The fractional Brownian sheet can be also defined by a Wiener integral with respect to the Brownian sheet (see [4])

W_s,t^α,β = _s

0

_t

0

Kα(s, u)Kβ(t, v)dWu,v. (3) whereα, β ∈ (0,1)and the kernelsKα, Kβ are defined above. Note that this process is a two-parameters centered Gaussian process, starting from(0,0), and its covariance is given by

E

Ws,t^α,βW_s^α,β,t

= 1 2

s²^α +s²^α − |s−s|²^α1 2

t²^β +t²^β− |t−t|²^β ,

(4) and it coincides in law with the process introduced in [10] or [1].

The aim of this work is to study the weak convergence of some continuous processes in the anisotropic Besov spaces to the fractional Brownian sheet. We will consider two types of approximations. First, we let

yε(s, t)= _t

0

_s

0

1 ε²

√xy(−1)^N(^x^ε^,^y^ε⁾dxdy, ε >0,

where{N(x, y), (x, y)∈R²₊}is a standard Poisson process in the plane. It was proved in [3] (using the result of Stroock [11] for the one-dimensional case) that this process converges in law in the space of continuous functions on[0,1]², denoted by_C([0,1]²), as ε tends to 0, to the ordinary Brownian sheet. Using this result and the representation (3) a natural way to obtain an approximation in law of the fractional Brownian sheet is to put

Xε(s, t)= _t

0

_s

0

Kα(s, u)Kβ(t, v)1 ε²

√uv(−1)^N(^u^ε^,^v^ε⁾dudv.

The weak convergence of the family of processesXεto the fractional Brownian sheetW^α,βin the space of continuous functions has been showed in [4]. In our first result we prove that the sequenceXεconverges also toW^α,βin the stronger topology of the anisotropic Besov spacelip^∗_p((α, β), b)). See the next Section

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for the definition of Besov spaces. To simplify the notation, we putn= _ε¹2 and we will consider the family of processes

Xⁿ(s, t)=n _s

0

_t

0

Kα(s, x)Kβ(t, y)√xy(−1)^Nⁿ^(x,y)dxdy. (5) In the last expressionNn(x, y):=N(x√

n, y√

n). Observe thatNnis a Poisson process with intensityn.

On the other hand, it has been proved in [5] that if(Bⁿ)n,(Cⁿ)nare two families of independent one dimensional Brownian motions then the process

Wⁿ(s, t)= 1

√n n j=1

B_s^jCt^j

converges weakly to the Brownian sheet in the two dimensional Besov space lip^∗_p((¹₂,¹₂), b)for anyp > _b². Our second approximation result is an extension of the result of [5]. That is, if we put

Zⁿ(s, t)= 1

√n n

j=1

B_s^j,αCt^j,β (6)

where(B^n,α)n,(C^n,β)n are two families of independent one dimensional fractional Brownian motions, then the processZⁿconverges in law, asn→ ∞, to the fractional Brownian sheetW^α,βin the spacelip_p^∗((α, β), b).

For the weak convergence in Besov spaces to the one dimensional fractional Brownian motion we refer to [9].

2 Anisotropic Besov spaces

We recall in this section some basic elements on the two-dimensional Besov spaces. We refer to [10] for a complete presentation on this subject.

LetT²= [0,1]²andD= {1,2}. For a functionf :T²→R,i ∈D,h∈R andei =(δi1, δi2)unit vectors, let

h,i =

f (z+hei)−f (z), if z, z+hei ∈T² 0, otherwise

denote the progressive difference off in the directionei. Let (h1,h2)f =h1,1◦h2,2f, for (h¹, h²)∈R².

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IfA= {i}is a subset ofDwith one element, then we put(h1,h2),Af =hi,if and(h1,h2),Af =f ifA=.

Now, forf ∈L^p(T²)if 1≤p <∞orf ∈C(T²)(the space of continuous functions onT²) ifp= ∞, we define itsL^p -modulus of continuity by

ωp,A(f, (t1, t2))= sup

0≤h1≤t1,0≤h2≤t2

(h1,h2),Af pfor(t1, t2)∈R₊². Forbreal anda¯ =(a1, a2),a1, a2>0 consider the real valued application on T²given by

ωa,b¯ ((t1, t2), A)=

i∈A

t1^a¹

1+

i∈A

log1 ti

b

for anyA⊂Dwithωa,b¯ ((t1, t2), )=1.

Definition 2.1. Leta¯ = (a¹, a²) , a¹, a² > 0, b ∈ R and1 ≤ p ≤ ∞. The anisotropic Besov spaceLipp(a, b)¯ is defined by

Lipp(a, b)¯ =

f ∈L^p(T²);

A⊂D

sup

t1,t2>0

ωp,A(f, (t¹, t²)) ωa,b¯ ((t1, t2), A) <∞

and this space is endowed with the norm f ^a,b_p^¯ =

A⊂D

sup

t1,t2>0

ωp,A(f, (t1, t2)) ωa,b¯ ((t1, t2), A).

In this wayLipp(a, b)¯ becomes a non-separable Banach space.

We also introduce the subspacelip^∗_p(a, b)¯ ofLipp(a, b)¯ by

lip^∗_p(a, b)¯ =

f ∈L^p(T²); ∀=A⊂D, lim

δA(t1,t2)→0

ωp,A(f, (t¹, t²)) ωa,b¯ ((t1, t2), A) =0

whereδA(t1, t2)=min{ti, i∈A}.

The following result on Besov spaces and fractional Brownian motion has been proved in [10].

Theorem 2.1. For any2< p <∞, it holds P

W^α,β∈Lipp((α, β),0)

=1andP

W^α,β ∈lip^∗_p((α, β),0)

=0. For similar results in the one parameter case we refer to [7].

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3 Weak convergence to the fractional Brownian sheet in Besov spaces We will need the following tightness criterion in Besov spaces given by [5].

Lemma 3.1. Let(Uⁿ(s, t))_(s,t)∈[0,1]² be a sequence of continuous adapted pro- cesses such that there existsa = (a¹, a²), a¹, a² > 0 and for everyp ≥ 1it exists a constantCp>0such that, fors, s, t, t∈ [0,1],s ≤s, t ≤tit holds Es−s,t−tUⁿ(s, t)^p≤Cp|s−s|^a¹^p|t−t|^a²^p (7) and

Es−s,1Uⁿ(s, t)^p≤Cp|s−s|^a¹^p,

Et−t,2Uⁿ(s, t)^p ≤Cp|t−t|^a²^p. (8) ThenUⁿis tight in the spacelip^∗_p(a, b)for anyp >max_i ¹

ai ∨ ²_b.

We prove now the tightness of the approximating familiesXⁿ andZⁿ given by (5) and (6).

Lemma 3.2.

1) Let(Xⁿ(s, t))s,t∈[0,1]² be the family of processes given by (5). ThenXⁿis tight in the spacelip^∗_p((α, β), b))for anyp > ²_b∨ _α¹∨ _β¹.

2) Let(Zⁿ(s, t))s,t∈[0,1]² be the family of processes given by (6). ThenZⁿis tight in the spacelip^∗_p((α, β), b))for anyp > ²_b∨ _α¹∨ _β¹.

Proof. Tightness ofXⁿ: Note first that, since the Besov norms are increasing inpit suffices to prove the result forpeven. We will show that

sup

n E

s,tXⁿ(s, t) _p

≤Cp(s−s)^pα(t−t)^pβ

for anyp even. By Lemma 4.1 of [4], it suffices to check this inequality for s≥s >0,t≥t >0,s−s < sandt−t < t.

We will extend the kernelsKαandKβ over all(0,1]by putting K˜α(s, x)=

Kα(s, x) ifs > x

0 ifs ≤x

and for the sake of simplicity we will denote also byKα this extension. We introduce also the following notations

Kα,β(s, t, x, y)=Kα(s, x)Kβ(t, y),

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and

s,tKα,β(s, t, x, y)=(Kα(s, x)−Kα(s, x))(Kβ(t, y)−Kβ(t, y)).

We have that, with the notations introduced above, E

s,tXn(s, t)_p

=n^pE

[0,1]²s,tKα,β(s, t, x, y)√xy(−1)^Nⁿ^(x,y)dxdy _p

=n^pE

[0,1]^2p

p i=1

s,tKα,β(s, t, xi, yi)√

xiyi(−1)^Nⁿ^(xⁱ^,yⁱ⁾

dx¹· · ·dyp

. We obtain now a bound for the expectation of the random part of the last expression. First of all, we have that

(−1)^pⁱ⁼¹^Nⁿ^(xⁱ^,yⁱ⁾ =(−1)^pⁱ⁼¹^0,0^Nⁿ^(xⁱ^,yⁱ⁾, and that

p i=1

⁰,0Nn(xi, yi)= p

i=1

s,tNn(xi, yi)+ p

i=1

s,0Nn(xi, t)

+ p

i=1

⁰,tNn(s, yi)+ p Nn(s, t),

and hence

(−1)^pⁱ⁼¹^0,0^Nⁿ^(xⁱ^,yⁱ⁾=

(−1)^pⁱ⁼¹^s,t^Nⁿ^(xⁱ^,yⁱ⁾(−1)^pⁱ⁼¹^s,⁰^Nⁿ^(xⁱ^,t)(−1)^pⁱ⁼¹⁰^,t^Nⁿ^(s,yⁱ⁾.

Since for allxi, yi, xj, yk the three intervals((s, t), (xi, yi)], ((s,0), (xj, t)]

and ((0, t), (s, yk)] are disjoint sets, the three factors of the last product are independent random variables. Moreover, if we majorize the expectation of the first factor by 1 we will obtain

E

(−1)^pⁱ⁼¹⁰^,⁰^Nⁿ^(xⁱ^,yⁱ⁾

≤exp{−2nt[(x(p)−x(p−1))+ · · · +(x(2)−x(1))]}

×exp{−2ns[(y(p)−y(p−1))+ · · · +(y(2)−y(1))]},

wherex(1), . . . , x(p)are the variablesx1. . . , xpordered in increasing order.

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Using now the fact that 2t > tand 2s > s, the last expression can be bounded by

exp{−nt[(x_(p)−x(p−1))+ · · · +(x(2)−x(1))]}

×exp{−ns[(y(p)−y(p−1))+ · · · +(y(2)−y(1))]}

≤exp{−n[(x(p)−x(p−1))y(p−1)+ · · · +(x(2)−x(1))y(1)]}

×exp{−n[(y(p)−y(p−1))x(p−1)+ · · · +(y(2)−y(1))x(1)]}.

Therefore, E

_s,tX_n(s, t)_p

≤(p!)²n^p

[0,1]^2p

^p

i=1

_s,tK_α,β(s, t, x_i, y_i)√ x_iy_i

×exp{−n[(xp−xp−1)yp−1+ · · · +(x2−x1)y1]}

×exp{−n[(y_p−y_p−1)x_p−1+ · · · +(y2−y1)x1]}

×1_{x₁_{≤···≤x}_p_}1_{y₁_{≤···≤y}_p_}

dx1· · ·dyp

≤Cp

n²

[0,1]²

s,tKα,β(s, t, x1, y1)s,tKα,β(s, t, x2, y2)√

x1x2y1y2

×exp[−n(x2−x1)y1−n(y2−y1)x1]

1_{x₁_≤x₂_}1_{y₁_≤y₂_}dx1· · ·dy2

^p₂ .

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We now divide the region of integration in two parts: A = {x1 ≤ x² ≤ 2x1, y1≤y2≤2y1}andA^c.

The integral of expression (9) over the regionAcan be majorized by Cpn²

[0,1]⁴

s,tKα,β(s, t, x1, y1)s,tKα,β(s, t, x2, y2)x1y1

×exp{−2n[(x2−x¹)y¹+(y²−y¹)x¹]}1_{x₁_≤x₂_{, y}₁_≤y₂_}

dx¹· · ·dy²

≤ Cpn²

[0,1]⁴

Kα(s, x1)−Kα(s, x1) 2

Kβ(t, y2)−Kβ(t, y2) 2

x1y1

×exp{−2n[(x2−x¹)y¹+(y²−y¹)x¹]}1_{x₁_≤x₂_{, y}₁_≤y₂_}

dx¹· · ·dy² + Cpn²

[0,1]⁴

Kα(s, x²)−Kα(s, x²) 2

Kβ(t, y¹)−Kβ(t, y¹) 2

x¹y¹

×exp{−2n[(x2−x¹)y¹+(y²−y¹)x¹]}1_{x₁_≤x₂_{, y}₁_≤y₂_}

dx¹· · ·dy². These last two summands can be treated in a similar way. In the first one we first integrate with respect tox²and then with respect toy¹, and in the second

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we integrate with respectx1andy2. In both cases we obtain the bound Cp

[0,1]²(Kα(s, x)−Kα(s, x))²(Kβ(t, y)−Kβ(t, y))²dxdy ^p₂

= Cp

E

s,tW_s^α,β,t

2^p₂

=Cp(s−s)^αp(t−t)^βp.

We consider now the integral given in (9) over the regionA^c. This region is the union ofB1 = {x1 ≤ x2, y1 ≤ y2, x2 >2x1}andB2 = {x1 ≤ x2, y1 ≤ y2, y2>2y1}. We will deal with the integral overB2, the other one is analogous.

In this case we obtain the following inequalities:

2(y2−y1)x1+2(x2−x1)y1≥(y2−y1)x1+x1y1+2(x2−x1)y1

≥ 1

2(y2−y1)x1+x1y1+1

2(x2−x1)y1= 1

2(y2x1+y1x2).

Thus, the integral given in expression (9) over the regionB2can be bounded by

Cpn²

[0,1]⁴

2 i=1

s,tKα,β(s, t, xi, yi)√ xiyi

×exp[−n

2(y2x1+x2y1)]1_{x₁_≤x₂_{, y}₁_≤y₂_}dx1· · ·dy2

≤Cpn²

[0,1]⁴

Kα(s, x1)−Kα(s, x1) 2

Kβ(t, y1)−Kβ(t, y1) 2

x1y1

×exp[−n

2(y²x¹+x²y¹)]1_{x₁_≤x₂_{, y}₁_≤y₂_}dx¹· · ·dy² +

[0,1]⁴

Kα(s, x²)−Kα(s, x²) 2

Kβ(t, y²)−Kβ(t, y²) 2

x²y²

×exp[−n

2(y2x1+x2y1)]1_{x₁_≤x₂_{, y}₁_≤y₂_}dx1· · ·dy2

.

By integrating in the first summand of the last expression with respect tox2

andy2and in the second summand with respect tox1andy1, we obtain that the last expression is bounded by

Cp

[0,1]²(Kα(s, x)−Kα(s, x))²(Kβ(t, y)−Kβ(t, y))²dxdy ^p₂

=Cp

E

s,tW_s^α,β,t

2^p₂

=Cp(s−s)^αp(t−t)^βp.

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Using the same calculus we can prove that conditions (8) of Lemma 1 are also verified. This finishes the proof of tightness ofXⁿ.

Tightness ofZⁿ: Concerning 2), observe that the rectangular increments of the processZⁿcan be written as

s,tZⁿ_s,t = 1

√n n j=1

B_s^j,α −B_s^j,α C_t^j,β −C_t^j,β

and by decomposingB_s^j,α −Bs^j,α = ^B^s^j,α ^−B^s^j,α

E

B_s^j,α −Bs^j,α

2s−s^α and similarly for C_t^j,β −C_t^j,β, we get, with(ζi, ηi)i a double sequence of i.i.d. random variables with standard normal distribution,

Es−s,t−tZⁿ_s,t^p

≤ (s−s)^pα(t−t)^pβE



 1

√n n

j=1

ζiηi





p

≤ Cp(s−s)^pα(t−t)^pβE

1 n

n i=1

ζ_i² ^p₂

≤ Cp(s−s)^pα(t−t)^pβ1 n

n i=1

E|ζi|^p ≤Cp(s−s)^pα(t−t)^pβ

and similarly (8) can be checked.

We can state now our main result.

Theorem 3.3. LetXⁿ andZⁿ be the families definied by (5) and(6). Then the following weak convergences hold

Xⁿ→W^α,βinlip^∗_p((α, β), b)asn→ ∞ and

Zⁿ→W^α,βinlip_p^∗((α, β), b)asn→ ∞.

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Proof. We refer to [4] for the proof of the convergence of the finite dimensional distributions ofXⁿ to those ofW^α,β. Concerning the familyZⁿ, let N integer, a1, . . . , aN ∈ R and (s1, t1), . . . , (sN, tN) ∈ [0,1]². We must see that the random variables

N k=1

akZn(sk, tk)= 1

√n N k=1

ak

× n j=1

_s_k

0

Kα(sk, uk)dW_u^j_k

tk

0

Kβ(tk, vk)dW_v^j_k

converge in law, asntends to infinity to N

k=1

akWs^α,βk,tk = N k=1

_s_k

0

_t_k

0

Kα(sk, uk)Kβ(tk, vk)dWuk,vk.

To prove this convergence, we will prove the convergence of the associated characteristic functions. In the sequel we will consider the extension of the kernelKα(t, s)on[0,1]by putting zero ifs ≥t. For the sake of simplicity we will denoted this extension also byKα(t, s).

Let(γ^k,l)l be a sequence of elementary functions converging, for everysk, toKα(sk,·)inL²(T )asl goes to∞and for everytk, let(ρ^k,l)l a sequence of elementary functions converging inL²(T ), asl→ ∞, toKβ(tk,·).

Let us denote by Z_k,lⁿ = 1

√n n j=1

_s_k

0

γ^k,l(uk)dW_u^j_k

tk

0

ρ^k,l(vk)dW_v^j_k

and

Zk,l= _s_k

0

_t_k

0 Kα(sk, u)Kβ(tk, v)dWu,v. For everyλreal, we have the following bound

E

e^iλ^N^k=1^a^k^Zⁿ^(s^k^,t^k⁾ −E

e^iλ

_N

k=1akW_sk,tk^α,β

≤E

e^iλ^N^k=¹^a^k^Zⁿ^(s^k^,t^k⁾ −E

e^iλ

_N

k=1akZ_k,lⁿ +E

e^iλ^N^k=¹^a^k^Z^k,lⁿ −E

e^iλ^N^k=1^a^k^Z^k,l +E

e^iλ^N^k=1^a^k^Z^k,l −E

e^iλ

_N

k=1akW_sk,tk^α,β =I1+I2+I3

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By the mean value theorem we can bound the termI1by maxk EZⁿ(sk, tk)−Z_k,lⁿ and moreover

EZⁿ(sk, tk)−Z_k,lⁿ

≤ 1

√nE

n j=1

1

0 Kα(sk, uk)dW_u^j_k

1

0 Kβ(tk, vk)dW_v^j_k

− 1

0

γ^k,l(uk)dW_u^j_k

1 0

ρ^k,l(vk)dW_v^j_k

≤ 1

√nE n j=1

1 0

Kα(sk, u)−γ^k,l(u)

dW^j(u)

1 0

Kβ(tk, v)dW_v^j + 1

√nE n j=1

1 0

γ^k,l(u)dW_u^j

1 0

Kβ(tk, v)−ρ^k,l(u)

dW^j(v)

≤ 1

√n



E



ⁿ

j=1

1 0

Kα(sk, u)−γ^k,l(u)

dW^j(u)

1 0

Kβ(tk, v)dW_v^j









1 2

+ 1

√n



E



ⁿ

j=1

1 0

γ^k,l(u)dW^j(u)

1 0

Kβ(tk, v)−ρ^k,l(v) dW_v^j









1 2

. Using the independence of increments, we can majorize the last expression by

1 0

Kα(sk, u)−γ^k,l(u)2

du

1

0 K_β²(tk, v)dv

+ 1

0

(γ^k,l(u))²du

1 0

Kβ(tk, v)−ρ^k,l(v)2

dv

.

Now, sinceγ^k,landρ^k,l are elementary functions, the convergence ofI²to 0 as n→ ∞follows from the convergence ofWⁿtoW^α,β. Finally, concerning the last termI3, we have

E

e^iλ^N^k=¹^a^k^Z^k,l −E

e^iλ^N^k=¹^a^k^W^sk,tk^α,β ≤Cmax

k EZk,l−W^α,β(sk, tk)

=Cmax

j { ¯E|

[0,1]²[γ^kl(x)ρ^kl(y)−Kα(sk, x)Kβ(tk, y)]dWx,y|}

≤Cmax

j γ^kl⊗ρ^kl−Kα(sk,·)⊗Kβ(tk,·) _L²_([0,1]²).

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This last norm inL²([0,1]²)tends to zero, asl→ ∞, independently ofn. This

fact concludes the proof.

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Ciprian A. Tudor

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