We extend to random fields case, the results of Woyczynski, who proved Brunk's type strong law of large numbers (SLLNs) for -valued random vectors under geometric assumptions. Also, we give probabilistic requirements for above-mentioned SLLN, related to results obtained by Acosta as well as necessary and sufficient probabilistic conditions for the geometry of Banach space associated to the strong and weak law of large numbers for multidimensionally indexed random vectors.

1. Introduction

We will study the limiting behavior of multiple sums of random vectors indexed by lattice points, so called random fields. Such research has roots in statistical mechanics and arised in the context of ergodic theory. Almost 70 years ago, Wiener considered double sums over lattice points with applications to homogenous chaos. Many aspects of present investigations of models of critical phenomena in statistical physics, crystal physics or Euclidean quantum field theories involve multiple sums of random variables with multidimensional indices. Multiparameter processes arise in applied context such as brain data imaging, and so forth.

Let , be the positive integer -dimensional lattice points with coordinate wise partial ordering, . Points in are denoted by , or more explicitly and stands for the -tuple . Also for , we define and . The notation means that or equivalently .

Let —be a probability space, —a real separable Banach space, —a family of -valued random vectors and set If then stands for the Bochner integral. Let be a -valued net and . We say that strongly as if for any , there exists such that implies or shortly for any , “occurs finitely often” (see Smythe [1], Fazekas [2]). Furthermore, let be an increasing family of sub -algebras of , that is,Now, we will introduce definition of martingale (submartingale) for real-valued random fields, Smythe [3] (for more information see Merzbach [4]). Through this paper satisfies condition CI (conditional independence)where denotes the componentwise minimum of and A-adapted, integrable process is called martingale (submartingale) ifThe main aim of this paper is to prove a couple Brunk type strong laws of large numbers for independent -valued random fields. To prove this we would like to apply, among others, maximal inequalities for real-valued submartingale fields. Main results concerning maximal inequalities for random variables indexed by multidimensional indices are due to Cairoli [5], Gabriel [6], Klesov [7], Smythe [3], Shorack and Smythe [8], as well as Wichura [9]. In [5], Cairoli gave counterexample that the well-known following Doob maximal inequality for submartingalescannot be proved for a discrete-time random fields, utilizing “one dimensional” idea, as well as Hajek-Renyi-Chow inequality [10] and in consequence Chow or Brunk type strong law of large numbers. This problem motivated us to make an effort to give some new results for strong law of large numbers for random fields.

To get the above-mentioned result we will exploit the idea of maximal inequality introduced by Christofides and Serfling [11].

Theorem 1.1. Let be a submartingale, satisfies (1.3), and let be a nonincreasing array of nonnegative numbers. Then for where and if for some

Proof. In the multidimensional martingale case, Theorem 1.1 was proved by Christofides and Serfling [11, Theorem 2.2] using properties of submartingale fields, thus assertion of Theorem 1.1 is true.

The following remark concerns the technique of the proof of Theorem 1.1 in martingale case.

Remark 1.2. In the proof of Theorem 2.2 of [11], the authors construct the sets (see the algorithm in [11]) and say “An explicit expression of the sets in terms of the sets is possible to derive, but such formula is notationally messy and complicated.” It seems that in the proof, we can use the sets constructed as follows (in the case , for simplicity).
Let set and set We obtain the sets by changing the order of taking maximum. In this construction we used idea introduced by Zimmerman [12]. Similarly to the sets constructed by Christofides and Serfling, , are disjoint, and , respectively, measurable, andSuch construction gives a simple formula and is very intuitive.

2. The Main Results

We start from the following generalization of Theorem 1.1.

Theorem 2.1. Let be a submartingale, satisfies (1.3), and let be a nonincreasing array of nonnegative numbers. Then for and , where , and if for some

Proof. Assume without loss of generality that the sum on right-hand side of (2.1) has minimum for . Let us put and define disjoint partition of as follow:for .
It is easy to see that and for . Now, let us observe that we can apply Theorem 1.1 to the “cubes” , where . Thus we have

The next lemma is an equivalent version of the result obtained by Martikainen [13, page 435] of Kronecker lemma in multidimensional case. Let , then

Lemma 2.2. Let be natural numbers, with and families of increasing, positive numbers such that strongly as and Furthermore be an array of positive numbers such thatThenfor every

Proof. By applying the Martikainen lemma to

Lemma 2.3. Let be a real separable Banach space and and , then the following properties are equivalent.
(i) is R-type p.(ii) There exists positive constant C such that for every and for any family of independent random vectors in with ,

Proof. For (Woyczyński [14]) and since are independent, the lemma is also valid for

Theorem 2.4. Let , and be field of independent, zero-mean, -valued random vectors such that If is R-type p, then

Proof. Let , since are independent, satisfies (1.3) and is real, nonnegative submartingale. By the definition of strong convergence of elements of and “event occurs finitely (infinitely) often,” it is enough to prove for any . Let us observe, that by Theorem 2.1, Lemma 2.3 and Hölder's inequality for some constants , we getwhere
Now, it is enough to prove that appropriate multiple series is finite. Changing the order of summation and comparing to integrals, for some constant and for every , we haveThe above expression contains the following types of sums:where , and is any subset of
Now, by Lemma 2.2, (2.13) tends to as as well as (2.14) for . Hence, we haveWe complete the proof by taking the minimum over of both sides of (2.15), combined with (2.8) and (2.11).

For , we let obtain the following result.

Corollary 2.5. Let , and let be sequence of independent, zero-mean, -valued random vectors such that If is R-type p, then

Corollary 2.5 for is due to Woyczyński [14], which generalized results of Hoffman-Jørgensen, Pisier and Woyczyński [15] (), and results due to Brunk [16], Prohorov [17] ().

Example 2.6. Let and let be a field of random vectors fulfilling the assumptions of Theorem 2.4. For , we define 2-dimensional sector as follow:Assume that are uniformly bounded by constant and Hence by comparison to integrals, we havefor
Thus, the condition (2.8) of Theorem 2.4 is met and we have
In Theorem 2.7, we will give necessary and sufficient probabilistic condition for the geometry of Banach space associated with the above-mentioned strong law. In Theorem 2.12 we will replace geometric condition of Theorem 2.4 mentioned by probabilistic one to obtain SLLN (2.9).

Theorem 2.7. Let , . The following conditions are equivalent:
(i) is R-type p. (ii) For every there exists such that for any independent, -valued, zero-mean random vectors , For , the theorem is due to Woyczyński [14].

Proof. (i)(ii) Using Chebyshev inequality ,Lemma 2.3 and Hölder's inequality ,we have
(ii)(i) Let and let be an arbitrary sequence of independent random vectors in , with and . SetThen Thus, (i) follows directly from Theorem 3.1 of Woyczyński [14].

Combining Theorem 2.7 with the result of Rosalsky and Van Thanh [18, Theorem 3.1], we get the following corollary.

Corollary 2.8. Let , and let be a separable Banach space. If is family of independent, -valued, zero-mean random vectors, then the following conditions are equivalent.
(i) for every and , there exists such that for any vectors , (ii)For every random vectors , the condition implies that the SLLN holds.

Before we state the next theorem, we need more notations and present some useful lemmas. Let and , where is defined as 0 and

Lemma (Fazekas [2, Lemma 2.5]). Let be independent symmetric -valued random vectors. Assume that for all ,then SLLN (2.9) holds.

Lemma (Fazekas [2, Lemma 2.3] with ). Let be a field of independent, symmetric, -valued random vectors. Ifthen .

Lemma 2.11. For , there exists a positive constant such that for any separable Banach space and any finite set of independent -valued random vectors with for all , the following holds.
For , if , then it is possible to take
For ,

Proof. For the result is due to de Acosta [19, Theorem 2.1]. Since are independent the theorem is true in the multidimensional case.

Theorem 2.12. Let be a family of independent -valued zero-mean, random vectors and assume strongly in probability. Then
(i) if , then implies SLLN (2.9),(ii)if , then implies SLLN (2.9).

Theorem 2.12 is multiple sum analogue of a strong law of large numbers, Theorem 3.2 of de Acosta [19].

Proof. (i) Let us assume that are symmetric (desymmetryzation is standard) and putBy assumption, it follows that and by the Borell-Cantelli lemma, it is enough to prove It follows from assumption thatthus by Lemma 2.10and on the virtue of Lemma 2.9 and the Borell-Cantelli lemma, the proof will be completed if we show that for any , Now, for any by Chebyshev inequality and Lemma 2.11, we have
(ii) The same arguments and Hölder's inequality


The author is grateful to the referees for carefully reading the manuscript and valuable comments which help make this paper more clear and led to essential improvement of the first version.