Almost-sure path properties of fractional brownian sheet

www.elsevier.com/locate/anihpb

Almost-sure path properties of fractional Brownian sheet

Wensheng Wang

Department of Statistics, East China Normal University, Shanghai 200062, China and

Department of Mathematics, Hangzhou Teachers College, Hangzhou 310036, China Received 6 October 2005; received in revised form 11 July 2006; accepted 12 September 2006

Available online 12 January 2007

Abstract

The almost sure sample function behavior of the vector-valued fractional Brownian sheet is investigated. In particular, the global and the local moduli of continuity of the sample functions are studied. These results give precise information about the continuity and the oscillation behavior of the sample functions.

©2007 Elsevier Masson SAS. All rights reserved.

Résumé

On étudie le comportement p.s. du mouvement brownien fractionnaire vectoriel, en particulier ses modules de continuité local et global.

©2007 Elsevier Masson SAS. All rights reserved.

Keywords:Gaussian process; Fractional Brownian sheet; Modulus of continuity; Law of the iterated logarithm

1. Introduction

The parameter space is R^N₊= [0,∞)^N; a typical parameter (“time points”) ist =(t₁, . . . , t_N), sometimes also written ast_i, orC, ift₁= · · · =t_N=C. Fors= s_i,t= t_i two points ofR^N₊ withs_i t_i, 1iN,[s, t] denotes aN-dimensionalintervalof the type

[s, t] = ×^N_i=1[s_i, t_i].

The N-dimensional Lebesgue measure is denoted by λ. Following Orey and Pruitt [9], we denote the Lebesgue measureλ([0, s][0, t])of the symmetric difference of two intervals [0, s],[0, t]byδ(s, t ), and simply by δ(t )in cases= 0:δ(t )=λ([0, t]).

The state space R^d, is endowed with the Euclidean norm · . Given a functionf:R^N₊ →R^d, the “increment f (Q)off over the intervalQ” is defined in just the same way as iff were a distribution function andf (Q)=

Qdf thef-measure ofQ, e.g.

f

s_i,t_i

=f (t₁, t₂)−f (t₁, s₂)−f (s₁, t₂)+f (s₁, s₂)

E-mail address:wswang@stat.ecnu.edu.cn.

1 The author would like to acknowledge the partial support from the Natural Science Foundation of China, grant No. 10401037.

0246-0203/$ – see front matter ©2007 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpb.2006.09.005

in caseN=2.

Now, to introduce the processes to be studied, letX^{(N )}:= {X^{(N )}_t , t∈R^N₊},N1, be aα-fractional Brownian sheet taking values inR¹, whereα=(α₁, . . . , α_N)∈(0,1)^N, i.e.X^{(N )}is a mean zero real-valued Gaussian process indexed byN-dimensional time withX_t=0 with probability one for alltlying on the boundary ofR^N₊, that is with δ(t )=0; whose covariance function is:

E

X_s^{(N )}X_t^{(N )}

=c_α,N N

i=1

|s_i|^2αi+ |t_i|^2αi− |s_i−t_i|^2αi

, (1.1)

wherec_α,N is a positive constant depending only onαandN. Ifα₁= · · · =α_N=1/2,X^{(N )}is usually referred as Brownian sheet orN-parameter Wiener process; we denote it byW^{(N )}or simplyW.α-fractional Brownian sheet is a natural generalization ofN-parameter Wiener process.

We denoteα-fractional Brownian sheet taking values inR^d,d1, byX^(N,d)or simplyX. In caseN=1,d=1, X is the standard fractional Brownian motion. Throughout this paper, without loss of generality, we assume the constant in (1.1)c_α,N=2^−Nso thatE[(X^{(N )}_t )²] =_N

i=1t_i^2αi.

TheN-parameter Wiener process W^{(N )} have been investigated by several authors. For more information about W^{(N )}we refer to Park [11], Pyke [12], Orey and Pruitt [9] and Ehm [5].

Recently, some sample-function behavior of theα-fractional Brownian sheet have been studied by several authors as natural generalization of Brownian sheet. Dunker [4], Mason and Shi [7] and Belinsky and Linde [1] investigated small ball probabilities forX^(N,1). Lu, Wang and Zhang [6] investigated large increment properties forX^(N,1)in case N=2, α1=α2.

In this paper we study the almost-sure sample path properties of the vector-valued fractional Brownian sheet with indexα=(α₁, . . . , α_N) (0< α_i<1, 1iN ). We show the global and the local continuity moduli of the sam- ple functions. These results give precise information about the continuity and the small fluctuations of the sample functions.

As convention, to prove the continuity moduli, we can follow the classic method (cf. Theorem 1.1.1 in Csörg˝o and Révész [3]) without too much difficulty. The main points that require some care is the choice of the manner of discretizing the unit time interval for various problems and solve some dependence-related problems. In order to solve dependence problems and to prove its existence we give a concrete representation ofα-fractional Brownian sheet (see Section 2). Such a representation is based on the idea of Talagrand’s (stochastic) integral representation for fractional Brownian motion (cf. Talagrand [13]. In fact, Talagrand [13] gave a representation for the fractional Lévy’s N-parameter Brownian motion, a different type of processes (Gaussian vector fields).)

The paper is organized as follows. In Section 2 we give several preliminary properties and results for the α-fractional Brownian sheet, which will be used in latter sections. The global and the local continuity moduli for X^(N,d)are discussed in Sections 3 and 4 respectively.

Throughout this paper, finite and positive constants whose values are unimportant will be denoted by c. For s, t∈R^N₊, st (resp. s < t) means that si ti (resp. si < ti) for all 1iN. For x ∈R, let logx =ln(x), log logx=ln(ln(x)).

2. Preliminaries

In this section we shall give several preliminary results for X that we need. We first state several useful facts forX^{(N )}(for intervals) theα-fractional Brownian sheet taking values inR¹. The first one is on the covariance function for intervals.

(i) For any intervals[s, t],[s, t] ⊂R^N₊ E

X^{(N )} [s, t]

X^{(N )} [s, t]

= 1 2^N

N

i=1

−|t_i−t_i|^2αi+ |t_i−s_i|^2αi+ |s_i−t_i|^2αi− |s_i−s_i|^2αi

. (2.1)

(ii) Self-similarity: The processes(_N

i=1C_i^−αi)X^{(N )}_C

iti andX^{(N )}_t

i,t_i ∈R^N₊, are identical in distribution, for every C_i ∈(0,∞)^N;X^{(N )}([s, t])is distributed like(_N

i=1(t_i−s_i)^αi)X^{(N )}₁ for every interval[s, t].

(iii) Stationary increments: The processes X^{(N )}([Ci +s,Ci +t])and X^{(N )}([s, t]),s, t∈R^N₊, are identical in distribution, for everyC_i ∈ [0,∞)^N.

(iv) Additivity: For any real number s, s₀, s such that ss₀s and any t, t∈R^N₊⁻¹, X^{(N )}([(s, t ), (s, t)])≡ X^{(N )}([(s, t ), (s₀, t)])+X^{(N )}([(s₀, t ), (s, t)]). In cases=s,X([(s, t ), (s, t)])≡0.

(v) The processes X^{(N )}([(0, t ), (1, t)])andX^(N−1)([t, t]),t, t∈R^N₊⁻¹, are identical in distribution, where the latter process is the(α₂, . . . , α_N)-fractional Brownian sheet.

Remark 2.1.For increments of the formX^{(N )}_t −X^{(N )}_s (iii) is completely false. In particular ifs= s_i,t= s_i+u_i, the varianceX^{(N )}_t −X^{(N )}_s will depend crucially ons.

Now we establish some Fernique type inequalities forX^(N,d). Let now, as well as in the rest paper, for brevity’s sake,X:=X^(N,d).

Lemma 2.1.For any0< ε <1there existsc=c(ε, α, N, d)such that

P sup

xitxi+Ti sup

0sai

X

[t, t+s]u N

i=1

a^α_iⁱ

c

_N

i=1

T_i a_i +1

e^−u2/(2+ε) (2.2) for everyuu₀>0,x_i ∈R^N₊, andT_i,a_i ∈R^N₊. Particularly, for any0< ε <1there existsc=c(ε, α, N, d) such that

P sup

R⊂[xi,xi+ai]

X(R)u N

i=1

a^α_iⁱ

ce^−u2/(2+ε) (2.3)

for everyuu0>0,xi,ai ∈R^N₊, whereR= [yi,zi]ranges over intervals.

Proof. We first show (2.2). By stationary increments for intervals (see (iii)), we can assume that x_i = 0. For notation simplicity only, we give the proof forN=1 and=2. The following fact will be used in the proof: LetU be a normal random variable inR^dwith mean zero and identity covariance matrix. Then

c1(d)u^d−2e^−u2/2P

Uu

c2(d)u^d−2e^−u2/2 (2.4)

for anyuu₀>0. (SinceU²has aχ²distribution withddegrees of freedom.)

In caseN=1, by (2.4), (2.2) is an immediate consequence of Lemma 2.4 of Csáki, Csörg˝o and Shao [2]. For case N=2, by using (iv) and (2.4), the proof is similar to that of Lemma 2.5 of Lu, Wang and Zhang [6]. Here we give a sketch of the proof for the sake of completeness. For any positive real numbert andrputt_r =a₁[t·(2^2r/a₁)]/2^2r. By (iv) we have

X

[x, x+s] × [y, y+t]X

x_r, (x+s)_r

× [y, y+t]+X

(x+s)_r, x+s

× [y, y+t] +X

[x_r, x] × [y, y+t]

X

x_r, (x+s)_r

× [y, y+t] +^∞

j=0

X

(x+s)_r+j, (x+s)_r+j+1

× [y, y+t]

+^∞

j=0

X

[x_r+j, x_r+j+1] × [y, y+t].

Using (2.4) and repeating the lines of the proof of Lemma 2.1 of Csáki, Csörg˝o and Shao [2], we have for any y, t∈R¹₊,

P

sup

0xT1

sup

0sa1

X

[x, x+s] × [y, y+t]ua₁^α1t^α2 c

T₁ a₁+1

e^−u2/(2+ε). (2.5)

By using (2.5) and proceeding to the same lines of the above proof for the second coordinate, we have P

sup

0xT₁ 0yT2

sup

0sa₁ 0ta2

X

[x, x+s] × [y, y+t]ua₁^α1a₂^α2 c

T₁ a₁ +1

T₂ a₂ +1

e^−u2/(2+ε).

This proves (2.2).

By choosingTi=ai for alliin (2.2), we get (2.3) immediately. The proof of Lemma 2.1 is complete. 2 We conclude this section by giving a concrete representation of the processX^{(N )}, which is very useful to show the existence ofX^{(N )}and to solve dependence problems involved in Section 4.

LetW^{(N )}(x), x∈R^N, be aN-parameter Wiener process. Then, we assert that the process X^{(N )}_t =

R^N

N

i=1

c(α_i)(1−cos(tix_i)+sin(tix_i))

|x_i|^αi+1/2 dW^{(N )}(x) (2.6)

is a version ofα-fractional Brownian sheet, where c(α_i)=

2

R

(1−cosx) dx

|x|^2αi+1 ₋1/2

.

(It is a positive and finite constant depending only on αi.) The definition of stochastic integral with respect to N-parameter Wiener process was given by Park [11].

In fact, such a representation is based upon the fact that if 0< H <1, there is a constantc(H )depending uponH only such that for eachtinR, by change of variable,

|t|^2H =c(H )²

R

2

1−cos(t x) dx

|x|^2H+1. To simplify notation, let

f (H, t, x)=c(H )(1−cos(t x)+sin(t x))

|x|^H+1/2 . (2.7)

Then

f (H, t, x)²=c(H )² 2

1−cos(t x) +2

1−cos(t x)

sin(t x)

/|x|^2H+1, and so

Rf (H, t, x)²dx= |t|^2H. Thus, using the relation f (H, t, x)f (H, s, x)=1

2

f (H, t, x)²+f (H, s, x)²−

f (H, t, x)−f (H, s, x)2 , it holds

R

f (H, t, x)f (H, s, x)dx=1 2

|t|^2H+ |s|^2H− |t−s|^2H . By Lemma 3 of Park [11] (see (4.15) on page 1588) it holds

E

X_t^{(N )}X_s^{(N )}

=

R^N

N

i=1

f (α_i, t_i, x_i)f (α_i, s_i, x_i)dx1· · ·dxN= N

i=1

R

f (α_i, t_i, x_i)f (α_i, s_i, x_i)dxi

=2^−N N

i=1

|t_i|^2H+ |s_i|^2H− |t_i−s_i|^2H . This verifies (1.1) and concludes the assertion.

3. Global moduli of continuity

In this section we shall show the global continuity moduli for theα-fractional Brownian sheetX:=X^(N,d). Theorem 3.1(Global continuity modulus for intervals). Let

d(s, t )= N

i=1

(t_i−s_i)^αi, λ [s, t]

= N

i=1

(t_i−s_i), β(s, t )=d(s, t )logλ

[s, t]^1/2, s, t∈R^N₊. Then

εlim→0 sup

s,t∈[0,1]^N, λ([s,t])ε

β(s, t )⁻¹X

[s, t]=2^1/2 a.s.

Remark 3.1.For a standard (one-dimensional) Wiener process, the form of the continuity modulus in Theorem 3.1 is slightly stronger than the classical Lévy’s form because, the normalized factor is√

|t−s||log|t−s||, while in the latter form that is√

ε|logε|.

Proof. At first, we show lim sup

ε→0

I (ε)2^1/2 a.s., (3.1)

where

I (ε)= sup

s,t∈[0,1]^N, λ([s,t])ε

β(s, t )⁻¹X [s, t].

Leti=(i1, . . . , i_N−1),m=(m1, . . . , m_N−1), and define the events forμ >0 E(i, j, k, m, n)=

supX

[s, t](1+μ)A

2 log B⁻¹ , where the supremum is taken over alls, tsatisfying

i_p

n2^−mp/nsp<i_p+1

n 2^−mp/n, p=1, . . . , N−1, i_p

n2^−mp/n+2^−(mp+1)/nt_p<i_p+1

n 2^−mp/n+2^−mp/n, p=1, . . . , N−1;

j−1

2ⁿ sN< j

2ⁿ, j+k

2ⁿ tN<j+k+1 2ⁿ , and

A=A(α, k, m, n)=

2^−1/n−n⁻¹α1+···+αN−1

2^{−(α1m1+···+αN−1mN−1)/n}

k2⁻ⁿαN

is the infimum ofd(s, t )for the intervals involved, and A=A(α, k, m, n)=

1+n⁻¹α1+···+αN−1

2^{−(α1m1+···+αN−1mN−1)/n}

(k+2)2⁻ⁿαN

is the corresponding supremum, and B=B(α, k, m, n)=

1+n⁻¹N−1

2^{−(m1+···+mN−1)/n}(k+2)2⁻ⁿ

is the supremum ofλ([s, t])for the intervals involved. The parameters will be restricted to the following ranges:

0i_pn2^mp/n, 1j2ⁿ, 1

4nkn, 0m_p< n², p=1, . . . , N−1,n=3,4, . . . .It is easy to check that

01−A A₋1

cn⁻¹→0.

By (2.3) (withu=(1+μ)A(A)⁻¹

2 log(B⁻¹)) we obtain P

E(i, j, k, m, n) cexp

−

1+μ 2

2

log B⁻¹ c2^{−(1+μ/4)2(m1+···+mN−1)/n}2^{−(1+μ/4)2n} for largen. Thus

n

i,j,k,m

P

E(i, j, k, m, n)

c

n

i,j,k,m

2^{−(1+μ/4)2(m1+···+mN−1)/n}2^{−(1+μ/4)2n}

c

n

m

n2^{−((1+μ/4)2−1)(m1+···+mN−1)/n}2^{−((1+μ/4)2−1)n}

c

∞ n=1

n^2N−12^{−((1+μ/4)2−1)n}<∞, where

i,j,k,m,

mtake over the above ranges ofi, j, k, m. Thus, by Borel–Cantelli lemma, a.s. only finitely many of the eventsE(i, j, k, m, n)occur. Then for a given sample path we can find an₀such that none of these occur for nn₀.

Let[s, t]be an interval witht_N−s_N=min{t_p−s_p}andλ([s, t])n^N₀2^{−N n0}. There is clearly no loss of generality in the first assumption; the second implies thatt_N−s_Nn₀2⁻ⁿ⁰. Now we choosenso that

(n+1)2⁻ⁿ⁻¹< tN−sNn2⁻ⁿ and thenj, k, m, andiso that

(j−1)2⁻ⁿsN< j2⁻ⁿ, (j+k)2⁻ⁿtN< (j+k+1)2⁻ⁿ,

2^−(mp+1)/n< t_p−s_p2^−mp/n, n⁻¹i_p2^−mp/ns_p< n⁻¹(i_p+1)2^−mp/n.

It is now easy to check that if[s, t] ⊂ [0,1]^Nthe restrictions on the indices are satisfied and[s, t]is one of the intervals in the eventE(i, j, k, m, n). This proves (3.1).

Next we show lim inf

ε→0 I (ε)2^1/2 a.s. (3.2)

Note that it is sufficient to prove it ford=1 sinceX([s, t])is larger than any of its component.

Letε_n=θ^−nN, θ >1, n1. Note that for anyεthere is an integernsuch thatε_n+1< εε_n. We have lim inf

ε→0 I (ε)lim inf

n→∞ inf

εn+1<εεn

I (ε) lim inf

n→∞ sup

s,t∈[0,1]^N, λ([s,t])ε_n+1

β(s, t )⁻¹X^{(N )} [s, t] lim inf

n→∞ max

1i_pθⁿ⁺¹/Kβ 0,

θ⁻⁽ⁿ⁺¹⁾₋1X^{(N )}

ipKθ⁻⁽ⁿ⁺¹⁾ ,

(ipK+1)θ⁻⁽ⁿ⁺¹⁾

=:J,

whereK >0 is a large constant which will be chosen later andf (ip)means aN-dimensional vector(f (i1), . . . , f (iN)).

For any real numberst, H, setf (t )=(1+t )^2H−t^2H. Then f(t )=2H

(1+t )^2H−1−t^2H−1 . It follows that

f(t )2H (2H−1)t^2H−2∨(1+t )^2H−2. Then, by fact (i), we have fori_p = j_p

E X^{(N )}

ipKθ⁻⁽ⁿ⁺¹⁾ ,

(ipK+1)θ⁻⁽ⁿ⁺¹⁾ X^{(N )}

jpKθ⁻⁽ⁿ⁺¹⁾ ,

(jpK+1)θ⁻⁽ⁿ⁺¹⁾

= 1

2^Nθ^−2(n+1)αp N

p=1

−2(ipK−j_pK)^2αp+(i_pK−j_pK+1)^2αp+(i_pK−j_pK−1)^2αp

= 1

2^Nθ^−2(n+1)αp N

p=1

f (i_pK−j_pK)−f (i_pK−j_pK−1)

1

2^Nθ^−2(n+1)

αp

N

p=1

f(ξp) ξp∈ [ipK−jpK−1, ipK−jpK] cθ^−2(n+1)αp

N

p=1

(i_pK−j_pK−1)^2αp−2∨(i_pK−j_pK+1)^2αp−2

cK^{max(2αp−2)}θ^−2(n+1)αp=:ρ(K)δ_n²,

whereρ(K):=cK^{max1pN(2αp−2)},δ_n:=θ^{−(n+1)Np=1αp}. Clearly, since 0< α_p<1,ρ(K)→0 asK→ ∞. Let τ, η_ip, i_p ∈Z^N₊, be independent mean zero Gaussian random variables with Eτ²=ρ(K)δ_n², Eη²_i

p= (1−ρ(K))δ_n². Letξi_p=X^{(N )}([ipKθ⁻⁽ⁿ⁺¹⁾,(ipK+1)θ⁻⁽ⁿ⁺¹⁾]). Then

E

(τ+η_ip)²

=δ_n²=E ξ²_i

p

and

E[ξ_ipξ_jp]ρ(K)δ²_n=E

(τ+η_ip)(τ+η_jp)

, i_p = j_p. Thus, by Slepian lemma, for anyμ >0

P max

1ipθⁿ⁺¹/Kξ_ip(1−μ)δ_n

2log(εn+1)!

P max

1ipθⁿ⁺¹/K(τ+η_ip)(1−μ)δ_n

2log(εn+1)!

P

1ipmaxθⁿ⁺¹/Kη_ip

1−μ 2

δ_n

2log(εn+1) +P

−τ μ

2

δ_n

2log(εn+1)

=

θⁿ⁺¹/K

i₁=1

· · ·

θⁿ⁺¹/K

i_N=1

P

η_ip

1−μ 2

δ_n

2log(εn+1) +P

−τ μ

2

δ_n

2log(εn+1)

. (3.3)

Noting the following well known fact: for allx >0 (2π )^−1/2

1−x⁻²

x⁻¹e^−x2/2P

N (0,1) > x

(2π )^−1/2x⁻¹e^−x2/2, whereN (0,1)is the standard normal random variable,

P

η_ip

1−μ 2

δ_n

2log(εn+1)

=P

N (0,1)

1−μ 2

2|log(εn+1)| 1−ρ(K)

=1−P

N (0,1) >

1−μ

2

2|log(εn+1)| 1−ρ(K)

1−ε⁽¹_n+⁻₁^{μ/4)2/(1−ρ(K))2}e^{−ε(1−μ/4)}

2/(1−ρ(K))2 n+1

for largen, where for obtaining the last inequality the following basic fact has been used:∀x, 1−xe^−x. Moreover P

−τμ 2δ_n

2log(εn+1)

=P

N (0,1)μ 2

2|log(εn+1)| ρ(K)

cε_n^μ₊^2/(4ρ(K)₁ ²⁾.

Thus we get that the right-hand side of (3.3) is bounded by e^{−θ(n+1)Nε(1−μ/4)}

2/(1−ρ(K))2

n+1 /K^N+cε_n^μ₊^2/(4ρ(K)₁ ²⁾.

Sinceρ(K)↓0, we can chooseK large enough such that(1−μ/4)²/(1−ρ(K))²<1. Thus the sum of the above bound is finite and hence the sum of the left-hand side of (3.3) is finite. By Borel–Cantelli lemma, we get

lim inf

n→∞ max

1i_pθⁿ⁺¹/K

ξi_p

δ_n

|log(εn+1)|2^1/2 a.s., which (also notingβ(0,θ⁻⁽ⁿ⁺¹⁾)=δn

|log(εn+1)|) impliesJ2^1/2a.s. This proves (3.2). The proof of Theo- rem 3.1 is complete. 2

Theorem 3.2(Global continuity modulus for points). Let h₁(ξ )=ξ^minαi

2|logξ|1/2

, h₂(ξ )=ξ^maxαi

2|logξ|1/2

, where the minimum and the maximum are taken over all1iN. Then

lim sup

ε→0

sup

s,t∈[0,1]^N, δ(s,t )ε

h1

δ(s, t )₋1

Xt−XsN^1/2 a.s., (3.4)

lim inf

ε→0 sup

s,t∈[0,1]^N, δ(s,t )ε

h2

δ(s, t )₋1

Xt−XsN^1/2 a.s. (3.5)

Remark 3.2.A special case is that ifα₁= · · · =α_N=H, (3.4) and (3.5) can be equalities. This special case is an extension of Theorem 2.4 of Orey and Pruitt [9] in fractional case.

Proof. The proof is obvious adaptation of the argument of Theorem 2.4 of Orey and Pruitt [9]. In order to show (3.4), in the proof of upper bound of Theorem 2.4 of Orey and Pruitt [9], it suffices to replace the use of Theorem 2.1 of Orey and Pruitt [9] by that of our Theorem 3.1, and replace the functionh(·)byh₁(·).

In order to show (3.5), in the proof of lower bound of Theorem 2.4 of Orey and Pruitt [9], it suffices to replace Amk, Bmj by the followingA_mk, B_mj respectively:

A_mk= 1

2+kK m η,0

,

1

2+kK+1 m η,1

2

, k=0, . . . , m−1;m=1,2, . . . , B_mj =

0,1

2+j K m η

,

1 2,1

2+j K+1

m η

, j=0, . . . , m−1;m=1,2, . . .

(where K is a large constant) and replace the functionh(·)by h2(·)because, following the second part proof of Theorem 3.1 (see the proof of (3.2)), (2.14) and (2.15) of Orey and Pruitt [9] hold withh(·)replaced by h₂(·)and A_mk, B_mj replaced byA_mk, B_mj respectively. This completes the proof. 2

4. Local moduli of continuity, laws of the iterated logarithm

In this section we shall show the local continuity moduli forX:=X^(N,d). Theorem 4.1(Local continuity modulus for intervals; LIL). Lets∈ [0,1]^N. Then

lim sup

ε→0

sup

t∈[0,1]^N, λ([s,t])ε

γ (s, t )⁻¹X

[s, t]=(2N )^1/2 a.s., (4.1)

where

γ (s, t )=d(s, t ) log log

λ

[s, t]₋11/2

, hereλ([s, t]), d(s, t )are defined as above. Particularly,

lim sup

ε→0

sup

t∈[0,1]^N, λ([0,t])ε

γ

0, t₋1

X_t =(2N )^1/2 a.s. (4.2)

Remark 4.1. WhenN =2, d =1, α1=α2=1/2, (4.2) is the well-known law of the iterated logarithm (LIL) of 2-parameter Wiener processes (cf. Theorem 1.12.2 in Csörg˝o and Révész [3]).

Proof. By fact (iii) (4.2) is equivalent to (4.1). So for convenience sake it is enough to show (4.2). We first show

the left-hand side of (4.2)(2N )^1/2 a.s. (4.3)

The proof can be given in a parallel way to the first part proof of Theorem 3.1. Letm=(m₁, . . . , m_N−1), and define the events forμ >0

F (k, m, n)=

supX_t(1+μ)A

2Nlog log B⁻¹ , where the supremum is taken over alltsatisfying

2^{−(mp+1)/logn}t_p<2^−mp/logn, p=1, . . . , N−1, k2⁻ⁿt_N< (k+1)2⁻ⁿ,

and

A=A(α, k, m, n)=2^{−(α1(m1+1)+···+αN−1(mN−1+1))/logn} k2⁻ⁿα_N

is the infimum ofd(0, t )for the intervals involved, and A=A(α, k, m, n)=2^{−(α1m1+···+αN−1mN−1)/logn}

(k+1)2⁻ⁿα_N

is the corresponding supremum, and

B=B(α, k, m, n)=2^{−(m1+···+mN−1)/logn}(k+1)2⁻ⁿ

is the supremum ofλ([0, t])for the intervals involved. The parameters are restricted to the following ranges:

1

4lognklogn, 0m_p< nlogn, n=3,4, . . . . It is easy to check that

01−A A₋1

c(logn)⁻¹→0.

By (2.3) (withu=(1+μ)A(A)⁻¹

2Nlog log(B⁻¹)),δ(t )=

it_i andB(logn+1)2⁻ⁿ, we obtain P

F (k, m, n) cexp

−

1+μ 2

2

Nlog log B⁻¹ cn^{−(1+μ/2)2N}

for largen. Thus

n

k,m

P

F (k, m, n)

c

∞ n=1

(logn)^Nn^{−(1+μ/2)2N+N−1}<∞.

Thus, by Borel–Cantelli lemma, a.s. only finitely many of theF (k, m, n)occur. Then for a given sample path we can find an₀such that none of these occur fornn₀.

Lett∈ [0,1]^N be a point witht_N=min{t_p}andδ(t )(logn₀)^N2^{−N n0}. There is clearly no loss of generality in the first assumption; the second implies thatt_N(logn0)2⁻ⁿ⁰. Now we choosenso that

log(n+1)

2⁻ⁿ⁻¹< t_N(logn)2⁻ⁿ and thenk, m, so that

k2⁻ⁿtN< (k+1)2⁻ⁿ, 2^{−(mp+1)/logn}< tp2^−mp/logn.

It is now easy to check that ift∈ [0,1]^N the restrictions on the indices are satisfied andt is one of the point in the eventF (k, m, n). This proves (4.3).