# Strong laws of large numbers for arrays of random variables and stable random fields

Strong laws of large numbers are established for random fields with weak or strong dependence. These limit theorems are applicable to random fields with heavy-tailed distributions including fractional stable random fields. The conditions for SLLN are described in terms of the p-th moments of the partial sums of the random fields, which are convenient to verify. The main technical tool in this paper is a maximal inequality for the moments of partial sums of random fields that extends the technique of Levental, Chobanyan and Salehi chobanyan-l-s for a sequence of random variables indexed by a one-parameter.

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## 1. Introduction

Many authors have studied strong laws of large numbers (SLLN) for arrays of random variables or, more generally, random fields with certain dependence structures. For example, Klesov [8, 9] proved a strong law of large numbers for orthogonal random fields and related asymptotic properties. Móricz [13, 14, 15, 16, 17, 18] established SLLN for quasi-orthogonal or quasi-stationary random fields. Móricz, Stadtmüller and Thalmaier [19] proved SLLN for blockwise -dependent random fields under moment conditions. Thanh [28] provided a necessary and sufficient condition for a general -dimensional arrays of random variables to satisfy a strong law of large numbers, which can be applied to blockwise independent random fields.

For random fields with more information on their dependence structures such as linear random fields, more results on their limiting behaviors have been known. For example, Marinucci and Poghosyan [12], and Paulauskas [22]

studied the asymptotics for linear random fields, including law of large numbers, central limit theorems and invariance principles, by applying the Beveridge-Nelson decomposition method. By applying ergodic theory, Banys et al.

[2] studied strong law of large numbers of linear random fields generated by ergodic or mixing random variables. Recently, Sang and Xiao [26] proved exact moderate and large deviation results for linear random fields with independent innovations and applied them to establish laws of the iterated logarithm for linear random fields.

This paper is mainly motivated by our interest in asymptotic properties of fractional random fields with long-range dependence and/or heavy-tailed distributions. An important class of such random fields is formed by fractional stable fields (cf. e.g., Samorodnitsky and Taqqu [25], Cohen and Istas [7], Pipiras and Taqqu [23], Ayache, Roueff and Xiao [1], Xiao [30]).

We start with some notation and definitions. Let be the set of non-negative integers. For any , we write it as or . The -norm and -norm of are defined by

 ∥n∥=(n21+⋯+n2d)1/2,|n|=max{n1,n2,⋯,nd},

respectively. For any constant , we denote .

There is a natural partial order in : if and only if for all . For any , we denote , which is called an interval (or a rectangle) in .

Let be a real-valued random field indexed by the lattice and let be a sequence of increasing subsets of . Denote the partial sum of over by

 S(Γn)=∑k∈Γnξk.

Let be a function such that as We say that a random field satisfies the strong law of large numbers with respect to and if

 (1.1) lim|n|→∞S(Γn)φ(n)=0 a.s.

In the random field setting, the index sets can have various configurations, ranging from spherical to rectangular, and to non-standard shapes, which arise naturally in many applied areas such as spatial statistics. When is clear from the context, we will simply refer to (1.1) as a -SLLN.

In this paper, we will consider spherical and rectangular sets , see below. Móricz [13] considered SLLN over more types of increasing domains in .

Two types of functions are of particular interest in this paper:

• , where is a non-decreasing function such that as . Such functions are useful in establishing SLLN for (approximately) isotropic random fields, including a class of stable random fields with stationary increments in [30].

• , where are non-decreasing functions on such that as . Functions of this form arise naturally in studying SLLN for anisotropic random fields with multiplicative kernels. Typical examples are linear or harmonizable fractional stable sheets ([1, 30]).

Throughout this paper, is a constant. We first consider SLLN for partial sums over spherical domains , where for any ,

 Qr={k=(k1,⋯,kd)∈Nd0: ∥k∥≤r}.

If we replace the -norm by the -norm , then is the cube .

Let . We denote

 (1.2) S(m;n)=∑k∈Qn+m∖Qmξk.

The following SLLN holds under both and -norms. It can be applied to isotropic random fields.

###### Theorem 1.1.

Let be a non-decreasing function such that as . Assume that there are constants such that for all . Let be an integer that satisfies . If

 (1.3) ∑n∈N0supm∈N0E(|S(m;an)|pf(an)p)<∞,

then

 (1.4) limn→∞S(0;n)f(n)=0,  a.s.

The second theorem proves a SLLN for partial sums over rectangular domains , that can be applied to anisotropic random fields.

For any , denote by

 (1.5) S(m;n)=m1+n1∑k1=m1+1⋯md+nd∑kd=md+1ξ(k1,⋯,kd),

the partial sum of the random variables over the interval .

###### Theorem 1.2.

Suppose that, for , is an increasing function that satisfy for all for some constants . Assume is an integer that satisfies . If

 (1.6) ∑n∈Nd0supm∈Nd0E(|S(m;an1,⋯,and)|pφ1(an1)p⋯φd(and)p)<∞,

then satisfies the SLLN (1.1) with and .

The following direct consequence of Theorem 1.2 is convenient to use.

###### Corollary 1.3.

Let the functions () and the constant be as in Theorem 1.2. If there is a function such that

 E(|S(m;n)|p)≤g(n), for all  m,n∈Nd0

and

 ∑n∈Nd0g(an1,⋯,and)φ1(an1)p⋯φd(and)p<∞,

then satisfies the SLLN as in Theorem 1.2.

The method for proving Theorem 1.2 is different from that based on the Rademacher-Menshov-type maximal moment inequalities (e.g., [8, 9, 13, 14, 15, 16, 17, 18, 19, 28]). We rely more on the approach of Chobanyan et al. [6, 11] and Nane et al. [20]. In particular, we extend their maximal moment inequalities for sequences of random variables to the case of random fields. We should mention that it is more convenient to verify condition (1.6) when the random field has certain kind of stationarity (see Section 4).

The rest of this paper is organized as follows. In Section 2 and 3, we prove Theorems 1.1 and 1.2, respectively. Some applications of these theorems are given in Section 4, where various random fields including fractional stable random fields, orthogonal, and quasi-stationary random fields are considered.

## 2. Spherical sums: back to the one-dimensional case

In this section we prove Theorem 1.1.

###### Proof of Theorem 1.1.

Recall in (1.2), and let We divide the proof into two steps.

Step 1: Our first task is to establish a recursion for . We first consider the case . For any , notice that

 M(k;an+1)≤max{M(k;(a−1)an),|S(k;(a−1)an)|+M(k+(a−1)an;an)}

and in general for ,

 M(k;lan)≤max{M(k;(l−1)an),|S(k;(l−1)an)|+M(k+(l−1)an;an)}.

By using the elementary inequality we get

 (2.1) Mp(k;lan)≤max{Mp(k;(l−1)an),2p−1(|S(k;(l−1)an)|p+Mp(k+(l−1)an;an))}≤(2p−1−1)|S(k;(l−1)an)|p+Mp(k;(l−1)an)+2p−1Mp(k+(l−1)an;an).

Eq. (2.1) can be written as

 (2.2) Mp(k;lan)−|S(k;lan)|p≤Mp(k;(l−1)an)−|S(k,(l−1)an)|p+2p−1(Mp(k+(l−1)an;an)−|S(k+(l−1)an;an)|p)−|S(k;lan)|p+2p−1|S(k;(l−1)an)|p+2p−1|S(k+(l−1)an;an)|p.

For , define

 (2.3) Fl(n)=supk∈N0E(Mp(k;lan)−|S(k;lan)|pf(lan)p)   and   F(n+1):=Fa(n).

Here

 (2.4) F(0)=supk∈N0E(Mp(k;1)−|S(k;1)|pf(1)p)

By the definition of , in both -norm and -norm. Hence .

Dividing both sides of (2.2) by , taking expectations, and then the supremum over all the ’s, we get

 (2.5) Fl(n)≤f((l−1)an)pf(lan)pFl−1(n)+2p−1f(an)pf(lan)pF1(n)+Gl(n),

where is

 Gl(n)=supk∈N0{2p−1f(an)pf(lan)pE(∣∣∣S(k+(l−1)an,an)f(an)∣∣∣p)+2p−1f((l−1)an)pf(lan)pE(∣∣∣S(k,(l−1)an)f((l−1)an)∣∣∣p)−E(∣∣∣S(k;lan)f(lan)∣∣∣p)}.

The terms in contribute nicely in the recursion for because of the following observation. By the assumption, we have

 f(an)pf(an+1)p(1+(a−1)2p−1)≤(1+(a−1)2p−1)Cp⌊log2a⌋1:=c<1.

Let . By iterating (2.5) as in the proof of Theorem 2.1 in [20] we obtain the following recursion:

 (2.6)

Step 2: We use the recursion obtained in Step 1 to finish the proof. By summing up (2.6) from to and using (1.3) we get

 ∞∑n=1supk∈N0E(Mp(k;an)−|S(k;an)|pf(an)p)≤Da,p1−c∞∑n=0supk∈N0E∣∣S(k;an)f(an)∣∣p<∞.

Consequently, for , we have almost surely

 (2.7) Mp(lan;an)−|S(lan;an)|pf(an)p→0,   as n→∞.

Note that (1.3) implies that

 |S(lan;an)|pf(an)p→0,  as   n→∞.

It follows from this and (2.7) that almost surely

 (2.8) Mp(lan,an)f(an)p→0, as n→∞.

Since

 (2.9) infnf(an+1)f(an)≥C⌊log2a⌋1>1,

we have

 (2.10) M(an;(a−1)an)f(an)=f(an+1)−f(an)f(an)M(an;(a−1)an)f(an+1)−f(an)≥(C⌊log2a⌋1−1)M(an;(a−1)an)f(an+1)−f(an).

Notice that

 M(an;(a−1)an)≤M(an;an)+M(2an;an)+⋯+M((a−1)an;an),

we derive from (2.8) and (2.10) that almost surely

 (2.11) M(an;(a−1)an)f(an+1)−f(an)→0, as n→∞.

Now by the assumption on ,

 f(an+1)f(an)≤C⌊log2a⌋2

and by using Theorem 9.1 in Chobanyan, Levental and Mandrekar [5], we see that (2.11) implies

 limn→∞S(0;n)f(n)=0a.s.

This finishes the proof of Theorem 1.1 in the case . The case follows similarly and we omit the details. ∎

## 3. Rectangular sums: the chaining method

In this section we prove Theorem 1.2. We first state a random field version of the Toeplitz Lemma from Móricz [13]. It will play a crucial role in the proof of Theorem 1.2.

###### Lemma 3.1.

Let be a set of non-negative numbers with the following two properties: There is a finite constant such that

 (3.1) supm∈Nd0∑k∈Nd0w(m;k)≤C

and

 (3.2) lim|m|→∞w(m;k)=0

for all . If is a sequence of real numbers such that

 (3.3) s(k)→0  as  |k|→∞,

then

 t(m)=∑k∈Nd0w(m;k)s(k)→0  as  |m|→∞.

For , define

 (3.4) Ms(m;n)=max1≤k1≤n1,⋯,1≤ks≤ns|S(m1,⋯,md;k1,⋯,ks,ns+1,⋯,nd)|,

and

 (3.5) Fs,s−1(n)=supm∈Nd0E(Mps(m;an)−Mps−1(m;an)φ1(an1)p⋯φd(and)p).

We call the -dimensional maximal sum of size , and the -dimensional maximal sum. We will prove several maximal moment inequalities. The first is for the case of .

###### Lemma 3.2.

Let and let () be the functions as in Theorem 1.2. There is a constant such that for ,

 (3.6) ∑n∈NdFs,s−1(n)≤11−κ1∑n∈N0supm∈Nd0E(Mps−1(m;an)φ1(an1)p⋯φd(and)p).
###### Proof.

We start by establishing a recursion for in for each . By definition (3.4) we see that for ,

 (3.7) Md(m;an1+1,⋯,and−1+1,land)≤Md(m;an1+1,⋯,and−1+1,(l−1)and)+Md(m1,…,md−1,md+(l−1)and;an1+1,⋯,and−1+1,and).

It follows from (3.7) and the elementary inequality that

 Mpd(m;an1+1,an2+1,⋯,land)≤Mpd(m;an1+1,an2+1,⋯,and−1+1,(l−1)and)+Mpd(m1,⋯,md−1,md+(l−1)and;an1+1,an2+1,⋯,and−1+1,and).

By iterating the above inequality, we derive

 (3.8) Mpd(m;an1+1,an2+1,⋯,and+1)≤Mpd(m1,⋯,md;an1+1,an2+1,⋯,and−1+1,and)   +Mpd(m1,⋯,md−1,md+and;an1+1,an2+1,⋯,and−1+1,and)   +Mpd(m1,⋯,md−1,md+2and;an1+1,an2+1,⋯,and−1+1,and)+⋯   +Mpd(m1,⋯,md−1,md+(a−1)and;an1+1,an2+1,⋯,and−1+1,and).

By adding and subtracting terms in (3.8), we get

 Mpd(m;an1+1,⋯,and+1)−Mpd−1(m;an1+1,⋯,and+1)≤−Mpd−1(m1,⋯,md;an1+1,⋯,and+1)+Mpd(m1,⋯,md;an1+1,⋯,and−1+1,and)−Mpd−1(m1,⋯,md;an1+1,⋯,and−1+1,and)+Mpd−1(m1,⋯,md;an1+1,⋯,and−1+1,and)+Mpd(m1,⋯,md−1,md+and;an1+1,⋯,and−1+1,and)−Mpd−1(m1,⋯,md−1,md+and;an1+1,⋯,and−1+1,and)+Mpd−1(m1,⋯,md−1,md+and;an1+1,⋯,and−1+1,and)+⋯++Mpd(m1,⋯,md−1,md+(a−1)and;an1+1,⋯,and−1+1,and)−Mpd−1(m1,⋯,md−1,md+(a−1)and;an1+1,⋯,and−1+1,and)+Mpd−1(m1,⋯,md−1,md+(a−1)and;an1+1,⋯,and−1+1,and).

Dividing both sides by , taking expectations and then the supremum over we get,

 Fd,d−1(n1+1,⋯,nd+1)≤aφd(and)pφd(and+1)pFd,d−1(n1+1,⋯,nd−1+1,nd)   +H(n1+1,⋯,nd−1+1,nd),

where

The terms in help us get the recursion we need. By the assumption on , we have

 aφd(and)pφd(and+1)p≤aCp⌊log2a⌋1,d:=kd<1.

Thus we get the recursion formula for :

 Fd,d−1(n1+1,⋯,nd+1)≤kdFd,d−1(n1+1,⋯,nd−1+1,nd)+supm∈Nd0E(Mpd−1(m1,⋯,md;an1+1,⋯,and)φ1(an1+1)p⋯φd(and)p).

Summing this recursion over , we get

 ∑n∈NdFd,d−1(n)≤11−kd∑n∈Nd0supm∈N0E(Mpd−1(m;an)φ1(an1)p⋯φd(and)p).

With essentially the same argument as above we get for that

 (3.9) ∑n∈NdFs,s−1(n)≤11−ks∑nNd0supm∈Nd0E(Mps−1(m;an)φ1(an1)p⋯φd(and)p).

Thus, we obtain (3.6) with . ∎

The next lemma establishes a maximal moment inequality for .

###### Lemma 3.3.

Let and let () be as in Theorem 1.2. There exist constants and such that for ,

 (3.10) ∑n∈NdFs,s−1(n)≤Da,p1−κ2∑n∈Nd0supm∈Nd0E(Mps−1(m;an)φ1(an1)p⋯φd(and)p).
###### Proof.

We start with establishing a recursion for in for each . Observe that for ,

 (3.11) Md(m;an1+1,⋯,and−1+1,land)≤max{Md(m;a