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Random Oper. Stoch. Equ. 2017; 25(3): 195–202

Research Article
Mohamed Ait Ouahra and Aissa Sghir*

Weak convergence in a class of anisotropic
Besov–Orlicz space
DOI: 10.1515/rose-2017-0013
Received December 10, 2016; accepted June 10, 2017

Abstract: The main result of this paper is a new tightness criterion in a class of anisotropic Besov–Orlicz
spaces. This result generalizes those obtained by Boufoussi and Lakhel [9] in a class of standard anisotropic
Besov spaces, and by Ait Ouahra, Kissami and Sghir [3] in a class of one-parameter Besov–Orlicz spaces.
Keywords: Besov–Orlicz space, tightness, weak convergence, Donsker type of invariance principle,
fractional Brownian sheet, symmetric stable process, local time
MSC 2010: 60B12, 60G52
||
Communicated by: Vyacheslav L. Girko

1 Introduction
In the theory of stochastic processes, the space of continuous functions is a classical framework for many
works on limit theorems. However, the recent developments in the theory of wavelets and their applications in
probability and statistics showed the need of using more sophisticated functions spaces like Besov spaces. For
the one-parameter case, Ciesielski, Kerkyacharian and Roynette [11] have showed, by using the techniques
of constructive approximation of functions, that Besov spaces are isomorphic to spaces of real sequences.
The case of anisotropic Besov spaces was treated by Kamont [12]. These characterizations make the Besov
topology easy to handle, and many applications have been given in stochastic calculus (see, for example,
[1, 3, 4, 6–9, 17, 18]).
Our aim in this paper is to establish a tightness criterion in a class of anisotropic Besov–Orlicz spaces.
Our result generalizes those obtained by Boufoussi and Lakhel [9] in a class of standard anisotropic Besov
spaces, and by Ait Ouahra, Kissami and Sghir [3] in a class of one-parameter Besov–Orlicz spaces. These will
be done by recalling notions on anisotropic Besov–Orlicz spaces. We use result of Kamont [12] to give the
characterization of these spaces in terms of the coefficients of the expansion of a continuous function with
respect to a basis which consists of tensor products of Schauder functions.
The remainder of this paper is organized as follows. In Section 2, we present some basic facts about
anisotropic Besov–Orlicz spaces. In Section 3, we establish a tightness criterion in a class of anisotropic
Besov–Orlicz spaces. Finally, in Section 4, as applications, firstly, we generalize the results obtained by Rosen
[16] in the space of continuous functions, and recently by Ouahhabi and Sghir [17] in a class of standard
anisotropic Besov spaces, where the fractional Brownian motion is obtained as a limit in law of linear local
times of symmetric stable processes. Secondly, we generalize the classical Donsker type of invariance principle, where the limit in law is the Brownian sheet. Finally,we generalize the approximation result of fractional

Mohamed Ait Ouahra: Laboratoire de Modélisation Stochastique et Déterministe et URAC 04, Faculté des Sciences,
Université Mohammed Premier Oujda, BP 717, Oujda, Morocco, e-mail: ouahra@gmail.com
*Corresponding author: Aissa Sghir: Équipe EDP et Calcul Scientifique, Faculté des Sciences, Université Moulay Ismail Meknés,
BP 11201, Zitoune, Morocco, e-mail: sghir.aissa@gmail.com

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196 | M. Ait Ouahra and A. Sghir, Weak convergence in a class of anisotropic Besov–Orlicz space

Brownian sheet given by Tudor [18] in a class of standard anisotropic Besov spaces, where the fractional
Brownian sheet is obtained as a limit in law of the partial sums of two sequences of real independent fractional Brownian motions.
Most of the estimates in the sequel contain unspecified finite positive constants. We use the same symbol
C for these constants, even when they vary from one line to the next.

2 Anisotropic Besov–Orlicz spaces
In this section, we will present a brief survey of anisotropic Besov–Orlicz spaces. For more details, we refer
the reader to [12], and to [11] for the one-parameter Besov–Orlicz spaces. We use the same notations given
in [9].
Let (Ω, Σ, μ) be a σ-finite measure space. We denote by L p (Ω), 1 ≤ p < +∞, the space of Lebesgue integrable real-valued functions f on Ω with exponent p, endowed with the norm
1
p

‖f ‖p = ( ∫|f( ⋅ )|p dμ( ⋅ )) .


By a Young function M, we mean a convex, continuous and non-decreasing function M : ℝ+ → ℝ+ satisfying
M(0) = 0,

lim

x→+∞

M(x)
= +∞ and
x

lim

x→0

M(x)
= 0.
x

β

In this paper we are interested in the Young function M β (x) = e|x| − 1, β ≥ 1. The Orlicz space LM β (dμ) (Ω)
corresponding to M β is the Banach space of real-valued measurable functions f on Ω, endowed with the
norm
󵄨󵄨 f( ⋅ ) 󵄨󵄨
󵄨󵄨) dμ( ⋅ ) ≤ 1}.
‖f ‖M β (dμ) = inf { ∫ M β (󵄨󵄨󵄨
󵄨 λ 󵄨󵄨
λ>0


In case of (Ω, Σ, P) being a probability space, the Orlicz norm becomes
󵄨󵄨 f 󵄨󵄨
‖f ‖M β (dP) = inf {𝔼(M β (󵄨󵄨󵄨 󵄨󵄨󵄨)) ≤ 1}.
󵄨λ󵄨
λ>0
The following equivalence norm in LM β (dμ) (Ω) (see [11]) will be used in the sequel:
‖f ‖M β (dμ) ∼ sup
p≥1

‖f ‖p
1

.

(1)



For any function f : [0, 1]2 → ℝ and any h ∈ ℝ, the progressive difference in direction x1 (resp. x2 ) is defined
by
∆ h,1 f(x1 , x2 ) = f(x1 + h, x2 ) − f(x1 , x2 ),
∆ h,2 f(x1 , x2 ) = f(x1 , x2 + h) − f(x1 , x2 ).
For any (h1 , h2 ) ∈ ℝ2 , we set
∆ h1 ,h2 f = ∆ h1 ,1 ∘ ∆ h2 ,2 f,

∆2h,i f = ∆ h,i ∘ ∆ h,i f,

i = 1, 2.

For any Borel function f : [0, 1]2 → ℝ such that f ∈ L M β ([0, 1]2 ), one can measure its smoothness by its
modulus of continuity computed in the L M β ([0, 1]2 ) norm. To this end, let us define, for any t ∈ [0, 1] and
(t1 , t2 ) ∈ [0, 1]2 , the following:
ω(1,0) (f, t) := sup ‖∆ h1 ,1 f ‖M β ,
|h1 |≤t

ω(0,1) (f, t) := sup ‖∆ h2 ,2 f ‖M β ,
|h2 |≤t

ω(1,1) (f, t1 , t2 ) :=

sup
|h1 |≤t1 ,|h2 |≤t2

‖∆ h1 ,h2 f ‖M β .

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M. Ait Ouahra and A. Sghir, Weak convergence in a class of anisotropic Besov–Orlicz space

|

197

Definition 2.1. Let 0 < α1 , α2 < 1 and ν ∈ ℝ. The anisotropic Besov–Orlicz space, denoted by LipM β (α1 , α2 , ν),
is a non-separable Banach space of real-valued continuous functions f on [0, 1]2 , endowed with the norm
ω

α1 ,α2

‖f ‖Mνβ

:= ‖f ‖M β + sup

ω(1,0) (f, t1 )

α1 ,α2
(t1 , 1)
0<t1 ≤1 ω ν

ω(1,0) (f, t2 )

+ sup

where
α ,α2

ων 1

+

α1 ,α2
(1, t2 )
0<t2 ≤1 ω ν

α

α

(t1 , t2 ) = t11 t22 (1 + log(

ω(1,1) (f, t1 , t2 )

sup
0<t1 ,t2 ≤1

α ,α2

ων 1

(t1 , t2 )

,

ν
1
)) .
t1 t2

We consider the separable Banach subspace of LipM β (α1 , α2 , ν), defined as
α ,α2

(t1 , 1))

as t1 → 0,

α ,α2

(1, t2 ))

as t2 → 0,

Lip∗M β (α1 , α2 , ν) := {f ∈ LipM β (α1 , α2 , ν) : ω(1,0) (f, t1 ) = o(ω ν 1

ω(0,1) (f, t2 ) = o(ω ν 1

α ,α
o(ω ν 1 2 (t1 , t2 ))

ω(1,1) (f, t1 , t2 ) =

as t1 ∧ t2 → 0},

where t1 ∧ t2 := min(t1 , t2 ).
In order to state our main result, we need the following characterization theorem with respect to the
coefficients of the expansion of a continuous function with respect to a basis which consists of tensor products
of Schauder functions. For this, we put
1

A1 (p, j, l)(f ) =

1

2−j( 2 −α1 + p )
1

p β (1 + j)ν

A (p, j, l )(f ) =
󸀠

2−j( 2 −α2 + p )
1
β

p (1 +

A(p, j, j )(f ) =

2j+1

1
p

[ ∑ |C n,l󸀠 (f )| ] ,
p

l󸀠 = 0, 1,

n=2j +1

1

1

󸀠

j)ν

l = 0, 1,

n=2j +1

1

1

2

1
p

2j+1

[ ∑ |C l,n (f )|p ] ,

󸀠 1

1

2−j( 2 −α1 + p ) 2−j ( 2 −α2 + p )
1

p β (1 + j + j󸀠 )ν

2j+1

2j

󸀠 +1

1
p

∑ |C n,n󸀠 (f )| ] ,
p

[ ∑

n=2j +1 n󸀠 =2j󸀠 +1

where C n,n󸀠 (f ) = C1n ∘ C2n (f ), with
1
2
{
{C n (f )(t) = C n (f( ⋅ , t)), C n (f )(t) = C n (f(t, ⋅ )),
{
{C n (g) = 2 2j (2g( 2k − 1 ) − g( 2k − 2 ) − g( 2k ))
2j+1
2j+1
2j+1
{

(g defined on [0, 1]).

Now we are able to state the characterization theorem of our studied space, and we refer to [12] for more
details and a proof.
Theorem 2.2. (1) We have the following equivalence norm in LipM β (α1 , α2 , ν):
ω

α1 ,α2

‖f ‖Mνβ

∼ max{|C l,l󸀠 (f )|, sup sup A1 (p, j, l)(f ), sup sup A2 (p, j, l󸀠 )(f ), sup sup A(p, j, j󸀠 )(f )}.
j≥0 p≥1

j≥0 p≥1

j,j󸀠 ≥0 p≥1

(2) The subspace Lip∗M β (α1 , α2 , ν) corresponds to the sequences (C n,n󸀠 (f )) such that
lim A1 (p, j, l) = lim A2 (p, j, l󸀠 ) =

j→+∞

j→+∞

lim

j∨j󸀠 →+∞

A(p, j, j󸀠 ) = 0,

where l, l󸀠 = 0, 1.

3 Tightness in anisotropic Besov–Orlicz spaces
A family F of probability measures on the general metric space S is said to be tight if for each positive ε, there
exists a compact set K such that P(K) > 1 − ε for all P in F. In this section, we will establish a new tightness

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198 | M. Ait Ouahra and A. Sghir, Weak convergence in a class of anisotropic Besov–Orlicz space
criterion in the class of anisotropic Besov–Orlicz space Lip∗M β (α1 , α2 , ν). Our technique is similar to that given
by Boufoussi and Lakhel [9] in a class of standard anisotropic Besov spaces, and by Ait Ouahra, Kissami
and Sghir [3] in a class of one-parameter Besov–Orlicz spaces. The key ingredients to prove our result are an
extension, in Orlicz space, of the well-known Kolmogorov–Riesz theorem (see [13, p. 100, Theorem 11.4]) and
a result of Marcus and Pisier [14] (see Lemma 3.4 below) which allows us to introduce an Orlicz probability
measure in Theorem 2.2. We begin by the following two results, and we refer to [9] and [3] for more details
about their proofs.
Lemma 3.1. Let ε > 0, 0 < α1 , α2 < 1 and ν > 0. We set
H ε (f, α1 , α2 , ν, M β ) = sup

0<t1 ≤ε

ω(1,0) (f, t1 )
α ,α2

ων 1

(t1 , 1)

ω(1,0) (f, t2 )

+ sup

0<t2 ≤ε

α ,α2

ων 1

(1, t2 )

+ sup

t1 ∧t2 ≤ε

ω(1,1) (f, t1 , t2 )
α ,α2

ων 1

(t1 , t2 )

.

Let A be the space of measurable functions f : [0, 1]2 → ℝ such that
ω

α1 ,α2

sup ‖f ‖Mνβ

< ∞ and

lim sup sup H ε (f, α1 , α2 , ν, M β ) = 0.
ε→0

f ∈A

Then A is relatively compact in

Lip∗M β (α1 ,

f ∈A

α2 , ν).

Lemma 3.2. Let 0 < α1 , α2 < 1, 0 < ν < ν󸀠 . Then LipM β (α1 , α2 , ν) is compactly embedded in Lip∗M β (α1 , α2 , ν󸀠 ).
Now we are ready to state and prove our main result.
Theorem 3.3. Let {X n (s, t) : (s, t) ∈ [0, 1]2 }n≥1 be a sequence of random fields satisfying the following:
(i) X n ( ⋅ , 0) = X n (0, ⋅ ) = x for some x ∈ ℝ.
(ii) There exists a constant 0 < C < +∞ such that
󵄩󵄩 n
󵄩
󵄩󵄩X (s, t) − X n (s󸀠 , t) − X n (s, t󸀠 ) + X n (s󸀠 , t󸀠 )󵄩󵄩󵄩M β ≤ C|s − s󸀠 |α1 |t − t󸀠 |α2 ,
where 0 < α1 , α2 < 1.
Then the sequence (X n )n≥1 is tight in Lip∗M β (α1 , α2 , ν) for all ν > 1.
We need the following lemma, which was given in [14]. It will be the key ingredient to prove Theorem 3.3.
Lemma 3.4. Let {Z(t) : t ∈ T}, T = {1, 2, . . . , N}, be a stochastic processes defined on (Ω, Σ, P) and satisfying
‖Z(t)‖M β (dP) ≤ d

for all t ∈ T.

Then, for all β and β󸀠 such that 1 ≤ β ≤ β󸀠 ≤ ∞, we have
1

𝔼‖Z(t)‖M β󸀠 (dμ) ≤ dC β (log(N)) β

− β1󸀠

,

where dμ is a probability measure on T and C β is a finite positive constant depending only on β.
Proof of Theorem 3.3. Notice that by condition (i), we have C0,0 (X n ( ⋅ )) = x, C1,1 (X n ( ⋅ )) = X n (1, 1) − X n (0, 1)
and C n,0 (X n ( ⋅ )) = C0,n (X n ( ⋅ )) = 0 for all n ≥ 1. To prove Theorem 3.3, we are going to prove that for any ν > 1,
there exists a constant 0 < C < ∞ such that for all n ≥ 1, λ > 0 and for any 1 < ν󸀠 < ν, we have
ω

α1 ,α2
󸀠

P{‖X n ( ⋅ )‖Mνβ

> λ} ≤

C
.
λ

The last inequality implies that for any ε > 0, there exists λ0 large enough such that
ω

α1 ,α2
󸀠

P{‖X n ( ⋅ )‖Mνβ

> λ0 } ≤ ε

for all n ≥ 1.

By virtue of the characterization of Theorem 2.2, it suffices to prove that
I = P{ max{|C0,0 (X n ( ⋅ ))|, |C1,1 (X n ( ⋅ ))|, sup sup A1 (p, j, l)(X n ( ⋅ )),
j≥0 p≥1

sup sup A (p, j, l )(X ( ⋅ )), sup sup A(p, j, j󸀠 )(X n ( ⋅ ))} > λ} ≤
2

j≥0 p≥1

󸀠

n

j,j󸀠 ≥0 p≥1

C
.
λ

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|

199

We deal with one of the terms in I, since the proofs of the other terms are similar. We have
J = P{ sup sup A(p, j, j󸀠 )(X n ( ⋅ )) > λ}
j,j󸀠 ≥0 p≥1

1

= P{ sup sup

1

󸀠 1

1

2−j( 2 −α1 + p ) 2−j ( 2 −α2 + p )
1

p β (1 + j + j󸀠 )ν

j,j󸀠 ≥0 p≥1

2j

2j+1

󸀠 +1

∑ |C

[ ∑

n,n󸀠

1
p

(X ( ⋅ ))| ] > λ}.
n

p

n=2j +1 n󸀠 =2j󸀠 +1
󸀠

In order to apply Lemma 3.4, for each j, j󸀠 ≥ 0, let Z j,j be a stochastic process and μ the probability measure
󸀠
defined on T = {1, . . . , 2j } × {1, . . . , 2j } as follows:
󸀠

Z j,j (k, k󸀠 ) = C n,n󸀠 (X n ( ⋅ ))

for all (k, k󸀠 ) ∈ T,
1

󸀠

μ(Z j,j = C n,n󸀠 (X n ( ⋅ ))) =

for all (k, k󸀠 ) ∈ T.

󸀠
2j+j

󸀠

We denote by 𝔼μ the expectation of Z j,j with respect to μ. Then
‖Z

j,j󸀠

‖p := (𝔼μ |Z

j,j󸀠 p

1
p

| ) =2

− j+jp

󸀠

2j

2j+1

󸀠 +1

∑ |C

[ ∑

n,n󸀠

1
p

(X ( ⋅ ))| ] ,
n

p

n=2j +1 n󸀠 =2j󸀠 +1

and by virtue of (1), we get
󸀠

‖Z

j,j󸀠

‖M β (dμ) ∼ sup

‖Z j,j ‖p

p≥1

1

= sup
p≥1



2−

j+j󸀠
p
1



2j+1

2j

󸀠 +1

1
p

∑ |C n,n󸀠 (X ( ⋅ ))| ] .
n

[ ∑

p

n=2j +1 n󸀠 =2j󸀠 +1

Therefore, applying the Chebyshev inequality, we get
1

󸀠 1

1

󸀠 1

󸀠
2−j( 2 −α1 ) 2−j ( 2 −α2 ) j,j󸀠
1
2−j( 2 −α1 ) 2−j ( 2 −α2 )
‖Z

(dμ)
>
λ}

𝔼‖Z j,j ‖M β (dμ).

M
β
󸀠
󸀠
󸀠
ν
󸀠
ν
λ j,j󸀠 ≥0
(1 + j + j )
(1 + j + j )
j,j󸀠 ≥0

J = P{ sup

Now, according to [17] and condition (ii), there exists a constant 0 < C < +∞ such that
‖2−

j+j󸀠
2

󸀠

C n,n󸀠 (X n ( ⋅ ))‖M β (dP) ≤ C2−jα1 −j α2 .

Therefore, Lemma 3.4 implies that there exists a constant 0 < C β < +∞ such that
󸀠

󸀠

𝔼‖Z j,j ‖M β (dμ) ≤ C β 2−jα1 −j α2 .
Finally, we deduce that
J≤

1
1
C
=

λ j,j󸀠 ≥0 (1 + j + j󸀠 )ν󸀠
λ

for all ν󸀠 > 1.

This completes the proof of Theorem 3.3.

4 Applications
4.1 Limit in law of linear local times of symmetric stable processes
Let W α,β denote the fractional Brownian sheet of parameters α, β ∈ (0, 1). It is a centered Gaussian process,
starting from (0, 0), with covariance function
𝔼(W α,β (s, t)W α,β (s󸀠 , t󸀠 )) =
The case α = β =

1
2

1 󸀠2α
1
(s + s2α − |s󸀠 − s|2α ) (t󸀠2β + t2β − |t󸀠 − t|2β ).
2
2

correspond to the Brownian sheet.

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200 | M. Ait Ouahra and A. Sghir, Weak convergence in a class of anisotropic Besov–Orlicz space
The one-dimensional fractional Brownian motion of Hurst parameter α ∈ (0, 1) is the unique continuous
centered Gaussian process, starting from zero, with covariance function
𝔼(W tα W sα ) =

1 2α
(t + s2α − |t − s|2α ).
2

The case α = 21 corresponds to the Brownian motion.
Let X := {X t : t ≥ 0} denote the real-valued symmetric stable process of index 1 < α ≤ 2, which is known
to have a jointly continuous local time {L(t, x) : t ≥ 0, x ∈ ℝ} (see [5, 10]). For α = 2, X is a Brownian motion.
The following lemma gives a regularity property of the local time as a random function of two variables, and
its proof can be found in [2].
Lemma 4.1. There exists a constant 0 < C < ∞ such that for all 0 ≤ t, s ≤ 1, x, y ∈ ℝ and any integer p ≥ 1, we
have
1
α−1
α−1
󵄩󵄩
󵄩󵄩
2p
󵄩󵄩L(t, x) − L(s, x) − L(t, y) + L(s, y)󵄩󵄩2p ≤ C((2p)!) |t − s| 2α |x − y| 2 .
Now we are ready to state and prove the first application of our main result.
Theorem 4.2. The sequence of processes
{

1
ε

α−1
2

(L(t, εx) − L(t, 0)) : (t, x) ∈ [0, 1]2 }

converges in law, as ε → 0, to the process
1

{2√c α W α−1, 2 (L(t, 0), x) : (t, x) ∈ [0, 1]2 }
in Lip∗M1 ( α−1
2α ,

α−1
2 ,

ν) for all ν > 1.

Proof. By [16, Theorem 1.2], we have the convergence of the finite-dimensional distributions (see [19] for the
Brownian motion case). The tightness follows easily from Lemma 4.1 and Theorem 3.3. It suffices to use (1)
1
and the fact that ((2p)!) 2p ≤ 2p. This inequality was used also by Ait Ouahra, Kissami and Sghir [3] for the
one-parameter Besov–Orlicz spaces.

4.2 A Donsker type of invariance principle
To prove our next result, we need the following lemma, and we refer to [4] for a proof.
Lemma 4.3. Let 0 < β ≤ 2 and let {X n : n ≥ 1} be an independent family of centered and identically distributed
random variables such that ‖X1 ‖M β (dP) < ∞. Then there exists a constant 0 < C < ∞ such that
󵄩󵄩 X1 + ⋅ ⋅ ⋅ + X n 󵄩󵄩
󵄩󵄩
󵄩󵄩
≤ C‖X1 ‖M β (dP) .
󵄩󵄩
󵄩󵄩
√n
󵄩M β (dP)
󵄩
Theorem 4.4. Let {Z k,l : k ≥ 1, l ≥ 1} be an independent family of bounded, centered and identically distributed random variables such that ‖Z1,1 ‖M2 (dP) < ∞. Then the sequence of processes
{X n (s, t) :=

1 [ns] [nt]
∑ ∑ Z k,l : (t, s) ∈ [0, 1]2 }
n k=1 l=1

converges weakly, as n → +∞, to the Brownian sheet {B(s, t) : (t, s) ∈ [0, 1]2 } in the separable Banach spaces
Lip∗M2 (α, 21 , 12 ) for all ν > 1.
Proof. To prove the convergence of finite-dimensional distributions, one can check easily that the covariance between X n (s, t) and X n (s󸀠 , t󸀠 ) converges, as n → +∞, to that between B(s, t) and B(s󸀠 , t󸀠 ), which is
min(s, s󸀠 ) × min(t, t󸀠 ). It remains to show the tightness by using Lemma 4.3 and Theorem 3.3.

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M. Ait Ouahra and A. Sghir, Weak convergence in a class of anisotropic Besov–Orlicz space

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201

Indeed, we have
󵄩
󵄩󵄩 n
󵄩󵄩X (s, t) − X n (s󸀠 , t) − X n (s, t󸀠 ) + X n (s󸀠 , t󸀠 )󵄩󵄩󵄩M2 (dP)
󵄩󵄩
󵄩󵄩
[ns]
[nt]
󵄩1
󵄩
= 󵄩󵄩󵄩󵄩

∑ Z k,l 󵄩󵄩󵄩󵄩
󵄩󵄩M2 (dP)
󵄩󵄩 n [ns󸀠 ]+1 [nt󸀠 ]+1
󵄩󵄩
󵄩󵄩󵄩 √|[ns] − [ns󸀠 ]| × |[nt] − [nt󸀠 ]| [ns] [nt]
Z k,l
󵄩󵄩
󵄩󵄩
= 󵄩󵄩󵄩󵄩
.


󵄩󵄩
󸀠
󸀠
n
󵄩󵄩

[ns󸀠 ]+1 [nt󸀠 ]+1 |[ns] − [ns ]| × |[nt] − [nt ]| 󵄩M2 (dP)
By Lemma 4.3, there exists a constant 0 < C < ∞ such that
󵄩󵄩 [ns] [nt]
󵄩󵄩
Z k,l
󵄩󵄩
󵄩󵄩
󵄩󵄩 ∑
󵄩󵄩
≤ C,

󵄩󵄩 󸀠
󵄩
󸀠
󸀠
󵄩 [ns ]+1 [nt󸀠 ]+1 √|[ns] − [ns ]| × |[nt] − [nt ]| 󵄩󵄩M2 (dP)
and for n large enough, we have
√|[ns] − [ns󸀠 ]| × |[nt] − [nt󸀠 ]|
1
1
≤ |s − s󸀠 | 2 |t − t󸀠 | 2 .
n
Therefore, by using Theorem 3.3, we deduce the tightness.

4.3 Weak convergence to the fractional Brownian sheet
Theorem 4.5. Let (B n,α )n and (C n,β )n be two families of independent one-dimensional fractional Brownian motions. Then the sequence of processes
W n (s, t) =

1 n j,α j,β
∑ Bs Ct
√n j=1

converges weakly, as n → +∞, to the fractional Brownian sheet W α,β in the separable Banach spaces
Lip∗M2 (α, β, ν) for all ν > 1.
Proof. The convergence of finite-dimensional distributions can be deduced from a result of Tudor [18]. It
remains to show the tightness by using Theorem 3.3. Notice that W n (0, t) = W n (s, 0) = 0. We have
󵄩󵄩 n
󵄩
󵄩󵄩W (s, t) − W n (s󸀠 , t) − W n (s, t󸀠 ) + W n (s󸀠 , t󸀠 )󵄩󵄩󵄩M2 (dP)
󵄩󵄩
󵄩󵄩
󵄩 1 n j,α j,β
j,α j,β
j,α j,β
j,α j,β 󵄩
= 󵄩󵄩󵄩󵄩
∑ (B s C t − B s󸀠 C t − B s C t󸀠 + B s󸀠 C t󸀠 )󵄩󵄩󵄩󵄩
󵄩󵄩 √n j=1
󵄩󵄩M2 (dP)
󵄩󵄩
󵄩󵄩
󵄩 1 n j,α
j,β
j,β 󵄩
j,α
= 󵄩󵄩󵄩󵄩
∑ (B s − B s󸀠 )(C t − C t󸀠 )󵄩󵄩󵄩󵄩
󵄩󵄩M2 (dP)
󵄩󵄩 √n j=1
j,β
j,β
j,α
j,α
󵄩󵄩
󵄩󵄩 1 n
B s − B s󸀠
C t − C t󸀠
󵄩
󸀠 α
󸀠 β󵄩
󵄩
󵄩
= |s − s | |t − t | 󵄩󵄩
)(
)󵄩󵄩󵄩󵄩
∑(
j,β
j,β 2 󵄩
󵄩󵄩 √n j=1 √𝔼(B j,α − B j,α )2
√𝔼(C t − C 󸀠 ) 󵄩M2 (dP)
s
s󸀠
t
󵄩󵄩󵄩 1 n
󵄩󵄩󵄩
= |s − s󸀠 |α |t − t󸀠 |β 󵄩󵄩󵄩󵄩
∑ ξ i η i 󵄩󵄩󵄩󵄩
󵄩󵄩M2 (dP)
󵄩󵄩 √n j=1
≤ C|s − s󸀠 |α |t − t󸀠 |β ,
where {ξ i , η i , i ≥ 1} is a double sequence of independent identically distributed random variables with common law N(0, 1), and the last estimation is obtained by using Lemma 4.3.
Remark 4.6. Our result generalizes also the case of the Brownian sheet obtained by Nualart [15] in the space
of continuous functions, and Boufoussi and Dozzi [6] in a class of standard anisotropic Besov spaces.

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202 | M. Ait Ouahra and A. Sghir, Weak convergence in a class of anisotropic Besov–Orlicz space
Acknowledgment: The authors would like to thank the anonymous referee for her/his careful reading of the
manuscript and useful comments.

References
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[13]
[14]
[15]
[16]
[17]
[18]
[19]

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