Strong Laws of Large Numbers for 𝔹-Valued Random Fields

Abstract

We extend to random fields case, the results of Woyczynski, who proved Brunk's type strong law of large numbers (SLLNs) for 𝔹-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 𝑁^𝑟, 𝑟≥1 be the positive integer 𝑟-dimensional lattice points with coordinate wise partial ordering, ≤. Points in 𝑁^𝑟 are denoted by 𝐦, 𝐧 or more explicitly 𝐧=(𝑛₁,𝑛₂,…,𝑛_𝑟) and 𝟏 stands for the 𝑟-tuple (1,…,1). Also for 𝐧=(𝑛₁,𝑛₂,…,𝑛_𝑟), we define |𝐧|=∏^𝑟_𝑖=1𝑛_𝑖 and (𝐧)={𝐤∶𝐤≤𝐧}. The notation 𝐧→∞ means that max_{1≤𝑖≤𝑟}𝑛_𝑖→∞ or equivalently |𝐧|→∞.

Let (Ω,𝔉,𝑃)—be a probability space, (𝔹,‖⋅‖)—a real separable Banach space, {𝑋_𝐤,𝐤∈𝑁^𝑟}—a family of 𝔹-valued random vectors and set 𝑆_𝐧=_𝐤≤𝐧𝑋_𝐤. (1.1) If 𝐸‖𝑋‖<∞, then 𝐸𝑋 stands for the Bochner integral. Let {𝑎_𝐤,𝐤∈𝑁^𝑟} be a 𝔹-valued net and 𝑎∈𝔹. We say that 𝑎_𝐧→𝑎 strongly as 𝐧→∞ if for any 𝜀>0, there exists 𝐍_𝜀∈𝑁^𝑟 such that 𝐧≰𝐍_𝜀 implies ‖𝑎_𝐧−𝑎‖<𝜀 or shortly for any 𝜀>0, ‖𝑎_𝐧−𝑎‖≥𝜀 “occurs finitely often” (see Smythe [1], Fazekas [2]). Furthermore, let {𝔉_𝐤,𝐤∈𝑁^𝑟} be an increasing family of sub 𝜎-algebras of 𝔉, that is,_𝐤≤𝐧𝔉_𝐤⊂𝔉_𝐧⊂𝔉. (1.2)Now, we will introduce definition of martingale (submartingale) for real-valued random fields, Smythe [3] (for more information see Merzbach [4]). Through this paper {𝔉_𝐤,𝐤∈𝑁^𝑟} satisfies condition CI (conditional independence)𝐸(𝐸(⋅∣𝔉_𝐦)∣𝔉_𝐧)=𝐸(⋅∣𝔉_𝐦⋀𝐧)a.s., (1.3)where 𝐦⋀𝐧 denotes the componentwise minimum of 𝐦 and 𝐧. A{𝔉_𝐤,𝐤∈𝑁^𝑟}-adapted, integrable process {𝑍_𝐤,𝐤∈𝑁^𝑟} is called martingale (submartingale) if𝐸(𝑍_𝐧∣𝔉_𝐦)=𝑍_𝐦⋀𝐧(≥𝑍_𝐦⋀𝐧)a.s.∀𝐦,𝐧∈𝑁^𝑟. (1.4)The main aim of this paper is to prove a couple Brunk type strong laws of large numbers for independent 𝔹-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 submartingales𝜆𝑃(max_𝐤≤𝐧𝑆_𝐤≥𝜆)≤𝐸𝑆⁺_𝐧 (1.5)cannot 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 {𝑌_𝐤,𝔉_𝐤,𝐤∈𝑁^𝑟} be a submartingale, {𝔉_𝐤,𝐤∈𝑁^𝑟} satisfies (1.3), and let {𝐶_𝐤,𝐤∈𝑁^𝑟} be a nonincreasing array of nonnegative numbers. Then for 𝜆>0,𝜆𝑃(sup_𝐤≤𝐧𝐶_𝐤𝑌_𝐤≥𝜆)≤min_{1≤𝑠≤𝑟}{_𝐤≤𝐧(𝐶_𝐤−𝐶_{𝐤;𝑠;𝑘𝑠+1})𝐸𝑌⁺_𝐤−_{𝑘𝑖,𝑖≠𝑠}𝐶_{𝐤;𝑠;𝑛𝑠[⋃𝑛𝑠𝑘𝑠=1𝐵𝑘1𝑟(𝑠),…,𝑘]𝑐}𝑌⁺_{𝐤;𝑠;𝑛𝑠}𝑑𝑃}≤min_{1≤𝑠≤𝑟}{_𝐤≤𝐧(𝐶_𝐤−𝐶_{𝐤;𝑠;𝑘𝑠+1})𝐸𝑌⁺_𝐤}, (1.6) where 𝐶_{𝐤;𝑠;𝛼}=𝐶_{𝑘1,…,𝑘𝑠−1,𝛼,𝑘𝑠+1,…,𝑘𝑟} and 𝐶_𝐤=0 if 𝑘_𝑖>𝑛_𝑖 for some 𝑖=1,2,…,𝑟.

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 𝐵_𝐤^(𝑖) (see the algorithm in [11]) and say “An explicit expression of the sets 𝐵_𝐤^(𝑖) in terms of the sets 𝐴_𝐤 is possible to derive, but such formula is notationally messy and complicated.” It seems that in the proof, we can use the sets ^0𝑥0𝐵^02𝑑𝑐_𝐤^(𝑖) constructed as follows (in the case 𝑟=2, for simplicity).
Let 𝐧=(𝑛₁,𝑛₂), set 𝑍_𝑖(𝜔)=sup_{1≤𝑗≤𝑛2}𝐶_𝑖𝑗𝑌_𝑖𝑗(𝜔),𝐼⁽¹⁾(𝜔)=inf_{1≤𝑖≤𝑛1}{𝑖∶𝑍_𝑖(𝜔)≥𝜆}(or𝑛₁+1ifnosuch𝑖exists𝐽),₍₁₎(𝜔)=inf_{1≤𝑗≤𝑛2}{𝑗∶𝐶_𝐼𝑗𝑌_𝐼𝑗(𝜔)≥𝜆}, (1.7) and set ^0𝑥0𝐵^{02𝑑𝑐(1)}_𝑖𝑗={𝐼⁽¹⁾(𝜔)=𝑖,𝐽⁽¹⁾(𝜔)=𝑗}. (1.8)We obtain the sets ^0𝑥0𝐵^02𝑑𝑐_𝐤⁽²⁾ 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, ^0𝑥0𝐵^02𝑑𝑐_𝐤⁽¹⁾, ^0𝑥0𝐵^02𝑑𝑐_𝐤⁽²⁾ are disjoint, 𝔉_𝑖𝑛2 and 𝔉_𝑛1𝑗, respectively, measurable, and_{1≤𝑖≤𝑛1,1≤𝑗≤𝑛2}^0𝑥0𝐵^{02𝑑𝑐(1)}_𝑖𝑗=𝐵_{1≤𝑖≤𝑛1,1≤𝑗≤𝑛2}^{0𝑥002𝑑𝑐(2)}_𝑖𝑗=[sup_{1≤𝑖≤𝑛1,1≤𝑗≤𝑛2}𝐶_𝑖𝑗𝑌_𝑖𝑗≥𝜆]. (1.9)Such 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 {𝑌_𝐤,𝔉_𝐤,𝐤∈𝑁^𝑟} be a submartingale, {𝔉_𝐤,𝐤∈𝑁^𝑟} satisfies (1.3), and let {𝐶_𝐤,𝐤∈𝑁^𝑟} be a nonincreasing array of nonnegative numbers. Then for 𝜆>0 and 𝐦≤𝐧, 𝜆𝑃(sup_{𝐤∈[(𝐧)⧵(𝐦)]}𝐶_𝐤𝑌_𝐤≥𝜆)≤min_{1≤𝑠≤𝑟𝐤∈[(𝐧)⧵(𝐦)]}(𝐶_𝐤−𝐶_{𝐤;𝑠;𝑘𝑠+1})𝐸𝑌⁺_𝐤, (2.1) where 𝐶_{𝐤;𝑠;𝛼}=𝐶_{𝑘1,…,𝑘𝑠−1,𝛼,𝑘𝑠+1,…,𝑘𝑟}, and 𝐶_𝑘=0 if 𝑘_𝑖>𝑛_𝑖 for some 𝑖=1,2,…,𝑟.

Proof. Assume without loss of generality that the sum on right-hand side of (2.1) has minimum for 𝑠₀=1. Let us put 𝐷=(𝐧)⧵(𝐦) and define disjoint partition of 𝐷 as follow:𝐷₁={𝐣=(𝑗₁,…,𝑗_𝑟)∶𝑚₁+1≤𝑗₁≤𝑛₁,1≤𝑗₂≤𝑚₂,…,1≤𝑗_𝑟≤𝑚_𝑟𝐷},_𝑖={𝐣=(𝑗₁,…𝑗_𝑟)∶1≤𝑗₁≤𝑛₁,1≤𝑗₂≤𝑛₂,…,𝑚_𝑖+1≤𝑗_𝑖≤𝑛_𝑖,1≤𝑗_𝑖+1≤𝑚_𝑖+1,…,1≤𝑗_𝑟≤𝑚_𝑟}, (2.2)for 𝑖=2,3,…,𝑟.
It is easy to see that ⋃^𝑟_𝑖=1𝐷_𝑖=𝐷 and 𝐷_𝑖∩𝐷_𝑗=∅ for 𝑖≠𝑗. Now, let us observe that we can apply Theorem 1.1 to the “cubes” {𝐤∈𝐍^𝐫∶𝐥≤𝐤≤𝐧}, where 𝟏≤𝐥<𝐧. Thus we have𝜆𝑃(sup_{𝐤∈[(𝐧)⧵(𝐦)]}𝐶_𝐤𝑌_𝐤≥𝜆)=𝜆𝑃(_{𝐤∈[(𝐧)⧵(𝐦)]}[𝐶_𝐤𝑌_𝐤≥𝜆])=𝜆𝑃(^𝑟_{𝑖=1𝐤∈𝐷𝑖}[𝐶_𝐤𝑌_𝐤≥𝜆])≤𝜆^𝑟_𝑖=1𝑃(_{𝐤∈𝐷𝑖}[𝐶_𝐤𝑌_𝐤≥𝜆])≤𝜆^𝑟_𝑖=1𝑃(sup_{𝐤∈𝐷𝑖}𝐶_𝐤𝑌_𝐤≤≥𝜆)_𝑟𝑖=1min_{1≤𝑠≤𝑟𝐤∈𝐷𝑖}{(𝐶_𝐤−𝐶_{𝐤;𝑠;𝑘𝑠+1})𝐸𝑌⁺_𝐤}≤min^𝑟_{1≤𝑠≤𝑟𝑖=1𝐤∈𝐷𝑖}{(𝐶_𝐤−𝐶_{𝐤;𝑠;𝑘𝑠+1})𝐸𝑌⁺_𝐤}≤min_{1≤𝑠≤𝑟𝐤∈[(𝐧)⧵(𝐦)]}{(𝐶_𝐤−𝐶_{𝐤;𝑠;𝑘𝑠+1})𝐸𝑌⁺_𝐤}. (2.3)

The next lemma is an equivalent version of the result obtained by Martikainen [13, page 435] of Kronecker lemma in multidimensional case. Let 𝐥=(𝑙₁,…,𝑙_𝑠), 𝐦=(𝑚₁,…,𝑚_𝑡) then 𝑁^𝑠+𝑡∋(𝐥,𝐦)∶=(𝑙₁,…,𝑙_𝑠,𝑚₁,…,𝑚_𝑡).

Lemma 2.2. Let 𝑠,𝑡≥1 be natural numbers, with 𝑠+𝑡=𝑟 and {𝑎_𝐥,𝐥∈𝑁^𝑠},{𝑏_𝐦,𝐦∈𝑁^𝑡} families of increasing, positive numbers such that 𝑎_𝐥→∞,𝑏_𝐦→∞ strongly as 𝐥 and 𝐦→∞. Furthermore {𝑥_(𝐥,𝐦),(𝐥,𝐦)∈𝑁^𝑟} be an array of positive numbers such that_{(𝐥,𝐦)∈𝑁𝑟}𝑥_(𝐥,𝐦)𝑎_𝐥𝑏_𝐦<∞. (2.4)Then1𝑎_{𝐍1(𝐥,𝐦)≤(𝐍1,𝐍2)}𝑥_(𝐥,𝐦)𝑏_𝐦→0stronglyas𝑁^𝑠∋𝐍₁→∞ (2.5)for every 𝐍₂∈𝑁^𝑡.

Proof. By applying the Martikainen lemma to𝑦_(𝐥,𝐍2)=_𝐦≤𝐍2𝑥_(𝐥,𝐦)𝑏_𝐦where𝐍₂∈𝑁^𝑡. (2.6)

Lemma 2.3. Let (𝔹,‖⋅‖) be a real separable Banach space and 1≤𝑝≤2 and 𝑞≥1, then the following properties are equivalent.
(i)𝔹 is R-type p.(ii) There exists positive constant C such that for every 𝐧∈𝑁^𝑟 and for any family {𝑋_𝐤,𝐤∈𝑁^𝑟} of independent random vectors in 𝔹 with 𝐸𝑋_𝐤=0, 𝐸‖_𝐤≤𝐧𝑋_𝐤‖^𝑞≤𝐶𝐸(_𝐤≤𝐧‖𝑋_𝐧‖^𝑝)^𝑞/𝑝. (2.7)

Proof. For 𝑟=1 (Woyczyński [14]) and since {𝑋_𝐤,𝐤∈𝑁^𝑟} are independent, the lemma is also valid for 𝑟>1.

Theorem 2.4. Let 1≤𝑝≤2, 𝑞>1 and {𝑋_𝐤,𝐤∈𝑁^𝑟} be field of independent, zero-mean, 𝔹-valued random vectors such that min_{1≤𝑠≤𝑟𝐤∈𝑁𝑟}𝐸‖𝑋_𝐤‖^𝑝𝑞𝑘_𝑠|𝐤|^{𝑝𝑞−𝑞}<∞. (2.8)If 𝔹 is R-type p, then𝑆_𝐧|𝐧|→0stronglya.s.as𝐧→∞. (2.9)

Proof. Let 𝔉_𝐧=𝜎(𝑋_𝐤,𝐤≤𝐧), since 𝑋_𝐤 are independent, {𝔉_𝐤,𝐤∈𝑁^𝑟} satisfies (1.3) and {‖𝑆_𝐤‖^𝑝𝑞,𝔉_𝐤,𝐤∈𝑁^𝑟} is real, nonnegative submartingale. By the definition of strong convergence of elements of 𝔹 and “event occurs finitely (infinitely) often,” it is enough to provelim_𝑁→∞𝑃(sup_𝐤≰𝐍‖𝑆_𝐤‖|𝐤|≥𝜆)=0where𝑁^𝑟∋𝐍=(𝑁,…,𝑁) (2.10) for any 𝜆>0. Let us observe, that by Theorem 2.1, Lemma 2.3 and Hölder's inequality for some constants 𝐶, we get𝜆^𝑝𝑞𝑃(sup_𝐤≰𝐍‖𝑆_𝐤‖|𝐤|≥𝜆)≤min_{1≤𝑠≤𝑟𝐤≰𝐍}(1|𝐤|^𝑝𝑞−1(|𝐤|^𝑝𝑞;𝑠;𝑘_𝑠+1))𝐸‖𝑆_𝐤‖^𝑝𝑞≤𝐶min_{1≤𝑠≤𝑟𝐤≰𝐍}(1|𝐤|^𝑝𝑞−1(|𝐤|^𝑝𝑞;𝑠;𝑘_𝑠+1))𝐸(_𝐣≤𝐤‖𝑋_𝐣‖^𝑝)^𝑞≤𝐶min_{1≤𝑠≤𝑟𝐤≰𝐍}(1|𝐤|^𝑝𝑞−1(𝐤^𝑝𝑞;𝑠;𝑘_𝑠+1))|𝐤|^𝑞−1_𝐣≤𝐤𝐸‖𝑋_𝐣‖^𝑝𝑞≤𝐶min_{1≤𝑠≤𝑟𝐤≰𝐍}1𝑘_𝑠|𝐤|^{𝑝𝑞−𝑞+1}_𝐣≤𝐤𝐸‖𝑋_𝐣‖^𝑝𝑞, (2.11)where (|𝐤|^𝛽;𝑠;𝛼)∶=𝑘^𝛽₁⋅,…,⋅𝑘^𝛽_𝑠−1𝛼^𝛽𝑘^𝛽_𝑠+1⋅,…,⋅𝑘^𝛽_𝑟.
Now, it is enough to prove that appropriate multiple series is finite. Changing the order of summation and comparing to integrals, for some constant 𝐶>0 and for every 𝑠,1≤𝑠≤𝑟, we have_𝐤≤𝐍1𝑘_𝑠|𝐤|^{𝑝𝑞−𝑞+1}_𝐣≤𝐤𝐸‖𝑋_𝐣‖^𝑝𝑞=_𝐤≤𝐍𝐸‖𝑋_𝐤‖^𝑝𝑞_{𝐤≤𝐣≤𝐍}1𝑗_𝑠|𝐣|^{𝑝𝑞−𝑞+1}≤𝐶_𝐤≤𝐍𝐸‖𝑋_𝐤‖^𝑝𝑞(1𝑁^{𝑝𝑞−𝑞+1}−1𝑘_𝑠^{𝑝𝑞−𝑞+1})^𝑟_{𝑖=1,𝑖≠𝑠}(1𝑁^{𝑝𝑞−𝑞}−1𝑘_𝑖^{𝑝𝑞−𝑞}). (2.12)The above expression contains the following types of sums:_𝐤≤𝐍𝐸‖𝑋_𝐤‖^𝑝𝑞𝑁^{𝑝𝑞−𝑞+1}𝑁^{𝑙(𝑝𝑞−𝑞)}(𝑘_𝑖1⋅…⋅𝑘_𝑖𝑚)^{𝑝𝑞−𝑞}_𝐤≤𝐍,𝐸‖𝑋_𝐤‖^𝑝𝑞𝑘_𝑠^{𝑝𝑞−𝑞+1}𝑁^{𝑙(𝑝𝑞−𝑞)}(𝑘_𝑖1⋅…⋅𝑘_𝑖𝑚)^{𝑝𝑞−𝑞}, (2.13)(2.14)where 𝑙,𝑚=0,1,…,𝑟−1,𝑙+𝑚=𝑟−1, and {𝑖₁,⋅…⋅,𝑖_𝑚} is any subset of {1,2,…,𝑟}⧵{𝑠}.
Now, by Lemma 2.2, (2.13) tends to 0 as 𝑁→∞ as well as (2.14) for 𝑙≠0. Hence, we have_{𝐤∈𝑁𝑟}1𝑘_𝑠|𝐤|^{𝑝𝑞−𝑞+1}_𝐣≤𝐤𝐸‖𝑋_𝐣‖^𝑝𝑞≤𝐶_{𝐤∈𝑁𝑟}𝐸‖𝑋_𝐤‖^𝑝𝑞𝑘_𝑠|𝐤|^{𝑝𝑞−𝑞}. (2.15)We complete the proof by taking the minimum over 𝑠∈{1,…,𝑟} of both sides of (2.15), combined with (2.8) and (2.11).

For 𝑟=1, we let obtain the following result.

Corollary 2.5. Let 1≤𝑝≤2, 𝑞>1 and let {𝑋_𝑘,𝑘∈𝑁} be sequence of independent, zero-mean, 𝔹-valued random vectors such that ^∞_𝑛=1𝐸‖𝑋_𝑛‖^𝑝𝑞𝑛^{𝑝𝑞−𝑞+1}<∞. (2.16)If 𝔹 is R-type p, then𝑆_𝑛𝑛→0a.s.asn→∞. (2.17)

Corollary 2.5 for 𝑞≥1 is due to Woyczyński [14], which generalized results of Hoffman-Jørgensen, Pisier and Woyczyński [15] (1≤𝑝≤2,𝑞=1), and results due to Brunk [16], Prohorov [17] (𝔹=𝐑,𝑝=2,𝑞≥1).

Example 2.6. Let 𝑟=2 and let {𝑋_𝐤,𝐤∈𝑁²} be a field of random vectors fulfilling the assumptions of Theorem 2.4. For 𝜃∈𝑅₊, we define 2-dimensional sector 𝑁²_𝜃 as follow:𝑁²_𝜃={(𝑘₁,𝑘₂)∈𝑁²∶1≤𝑘₂≤𝜃𝑘₁}. (2.18)Assume that {𝐸‖𝑋_𝐧‖^𝑝𝑞,𝐧∈𝑁²} are uniformly bounded by constant 𝑀 and 𝑝𝑞−𝑞>1/2. Hence by comparison to integrals, we have_{(𝑘1,𝑘2)∈𝑁2𝜃}𝐸‖𝑋_{(𝑘1,𝑘2)}‖^𝑝𝑞𝑘_𝑠(𝑘₁𝑘₂)^{𝑝𝑞−𝑞}≤𝑀_{(𝑘1,𝑘2)∈𝑁2𝜃}1𝑘_𝑠(𝑘₁𝑘₂)^{𝑝𝑞−𝑞}<∞, (2.19)for 𝑠=1,2.
Thus, the condition (2.8) of Theorem 2.4 is met and we have∑_𝑁²_{𝜃∋𝐤≤𝐧}𝑋_𝐤|𝐧|→0stronglya.s.as𝐧→∞. (2.20)
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 1≤𝑝≤2, 𝑞≥1. The following conditions are equivalent:
(i)𝔹 is R-type p. (ii) For every 𝜆>0 there exists 𝐶_𝜆 such that for any independent, 𝔹-valued, zero-mean random vectors {𝑋_𝐤,𝐤∈𝑁^𝑟}, _{𝐤∈𝑁𝑟}|𝐤|⁻¹𝑃(‖𝑆_𝐤‖|𝐤|≥𝜆)≤𝐶_{𝜆𝐤∈𝑁𝑟}𝐸‖𝑋_𝐤‖^𝑝𝑞|𝐤|^{𝑝𝑞−𝑞+1}. (2.21)For 𝑟=1, the theorem is due to Woyczyński [14].

Proof. (i)⇒(ii) Using Chebyshev inequality ,Lemma 2.3 and Hölder's inequality ,we have _{𝐤∈𝑁𝑟}|𝐤|⁻¹𝑃(‖𝑆_𝐤‖|𝐤|≥𝜆)≤𝜆^−𝑝𝑞_{𝐤∈𝑁𝑟}𝐸‖𝑆_𝐤‖^𝑝𝑞|𝐤|^𝑝𝑞+1≤𝐶𝜆^−𝑝𝑞_{𝐤∈𝑁𝑟}|𝐤|^{−𝑝𝑞+𝑞−2}_𝐢≤𝐤𝐸‖𝑋_𝐢‖^𝑝𝑞≤𝐶𝜆^−𝑝𝑞_{𝐤∈𝑁𝑟}𝐸‖𝑋_𝐤‖^𝑝𝑞_𝐢≥𝐤|𝐢|^{−𝑝𝑞+𝑞−2}≤𝐶𝜆^−𝑝𝑞_{𝐤∈𝑁𝑟}𝐸‖𝑋_𝐤‖^𝑝𝑞|𝐤|^{𝑝𝑞−𝑞+1}. (2.22)
(ii)⇒(i) Let 𝐧=(𝑛₁,…,𝑛_𝑟) and let {𝑌_𝑛1,𝑛₁≥1} be an arbitrary sequence of independent random vectors in 𝔹, with 𝐸𝑌_𝑛1=0 and 𝑇_𝑚=∑^𝑚_𝑛1=1𝑌_𝑛1. Set𝑋_𝐧=𝑌_𝑛1for(𝑛₁,𝑛₂,…,𝑛_𝑟)=(𝑛₁0,1,…,1),for(𝑛₁,𝑛₂,…,𝑛_𝑟)≠(𝑛₁,1,…,1). (2.23)Then ^∞_𝑛1=1𝑛₁⁻¹𝑃(‖𝑇_𝑛1‖𝑛₁≥𝜆)=_{𝐧∈𝑁𝑟}|𝐧|⁻¹𝑃(‖𝑆_𝐧‖|𝐧|≥𝜆)≤𝐶_{𝜆𝐧∈𝑁𝑟}𝐸‖𝑋_𝐧‖^𝑝𝑞|𝐧|^{𝑝𝑞−𝑞+1}=𝐶_𝜆^∞_𝑛1=1𝐸‖𝑌_𝑛1‖^𝑝𝑞𝑛₁^{𝑝𝑞−𝑞+1}. (2.24)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 1≤𝑝≤2,𝑞≥1, and let 𝔹 be a separable Banach space. If {𝑋_𝐤,𝐤∈𝑁^𝑟} is family of independent, 𝔹-valued, zero-mean random vectors, then the following conditions are equivalent.
(i) for every 𝑞≥1 and 𝜆>0, there exists 𝐶_𝜆 such that for any vectors {𝑋_𝐤,𝐤∈𝑁^𝑟}, _{𝐤∈𝑁𝑟}|𝐤|⁻¹𝑃(‖𝑆_𝐤‖|𝐤|≥𝜆)≤𝐶_{𝜆𝐤∈𝑁𝑟}𝐸‖𝑋_𝐤‖^𝑝𝑞|𝐤|^{𝑝𝑞−𝑞+1}. (2.25)(ii)For every random vectors {𝑋_𝐤,𝐤∈𝑁^𝑟}, the condition_{𝐤∈𝑁𝑟}𝐸‖𝑋_𝐤‖^𝑝|𝐤|^𝑝<∞ (2.26) implies that the SLLN holds.

Before we state the next theorem, we need more notations and present some useful lemmas. Let 𝑁₀=(0,1,2,…) and 2^𝐧=(2^𝑛1,…,2^𝑛𝑟), where 2⁻¹ is defined as 0 andfor𝐤<𝐧denote𝑆^𝐧_𝐤=_{𝐤<𝐣≤𝐧}𝑋_𝐣. (2.27)

Lemma (Fazekas [2, Lemma 2.5]). Let {𝑋_𝐤,𝐤∈𝑁^𝑟} be independent symmetric 𝔹-valued random vectors. Assume that for all 𝜆>0,_{𝐤∈𝑁𝑟0}𝑃(|‖𝑆^𝟐𝐤_{𝟐𝐤−𝟏}‖−𝐸‖𝑆^𝟐𝐤_{𝟐𝐤−𝟏}‖||𝟐^𝐤−𝟏|𝑆>𝜆)<∞,𝐸(^𝐧|𝐧|)→0stronglyas𝐧→∞, (2.28)then SLLN (2.9) holds.

Lemma (Fazekas [2, Lemma 2.3] with 𝑎_k=|k|). Let {𝑋_𝐤,𝐤∈𝑁^𝑟} be a field of independent, symmetric, 𝔹-valued random vectors. If‖𝑋_𝐤‖≤|𝐤|a.s.forevery𝐤∈𝑁^𝑟,𝑆_𝐧|𝐧|→0stronglyinprobability, (2.29)then 𝐸‖𝑆_𝐧‖/|𝐧|→0stronglyas𝐧→∞.

Lemma 2.11. For 𝑞≥1, there exists a positive constant 𝐶_𝑞 such that for any separable Banach space 𝔹 and any finite set {𝑋_𝐤,𝐤≤𝐧} of independent 𝔹-valued random vectors with 𝑋_𝐤∈𝐿^𝑞 for all 𝐤≤𝐧, the following holds.
For 1≤𝑞≤2, 𝐸|‖𝑆_𝐧‖−𝐸‖𝑆_𝐧‖|^𝑞≤𝐶_{𝑞𝐤≤𝐧}𝐸‖𝑋_𝐤‖^𝑞, (2.30)if 𝑞=2, then it is possible to take 𝐶₂=4.
For 𝑞>2, 𝐸|‖𝑆_𝐧‖−𝐸‖𝑆_𝐧‖|^𝑞≤𝐶_𝑞[(_𝐤≤𝐧𝐸‖𝑋_𝐤‖²)^𝑞/2+_𝐤≤𝐧𝐸‖𝑋_𝐤‖^𝑞]. (2.31)

Proof. For 𝑟=1, the result is due to de Acosta [19, Theorem 2.1]. Since {𝑋_𝐤,𝐤∈𝑁^𝑟} are independent the theorem is true in the multidimensional case.

Theorem 2.12. Let {𝑋_𝐤,𝐤∈𝑁^𝑟} be a family of independent 𝔹-valued zero-mean, random vectors and assume ‖𝑆_𝐧‖/|𝐧|→0 strongly in probability. Then
(i) if 1≤𝑞≤2, then ∑_𝐧∈𝑁^𝑟(𝐸‖𝑋_𝐧‖^𝑞/|𝐧|^𝑞)<∞ implies SLLN (2.9),(ii)if 2≤𝑞, then ∑_𝐧∈𝑁^𝑟(𝐸‖𝑋_𝐧‖^𝑞/|𝐧|^𝑞/2+1)<∞ 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 {𝑋_𝐤,𝐤∈𝑁^𝑟} are symmetric (desymmetryzation is standard) and put𝑌_𝐤=𝑋_𝐤𝐼(‖𝑋_𝐤‖≤|𝐤|),𝑇_𝐧=_{𝐧∈𝑁𝑟}𝑌_𝐤. (2.32)By assumption, it follows that ∑_𝐧∈𝑁^𝑟𝑃(‖𝑋_𝐤‖≥|𝐤|)<∞ and by the Borell-Cantelli lemma, it is enough to prove 𝑇_𝐧/|𝐧|→0stronglya.s. It follows from assumption that𝑇_𝐧|𝐧|→0stronglyinprobability, (2.33)thus by Lemma 2.10𝐸‖𝑇_𝐧‖|𝐧|→0stronglyas𝐧→∞, (2.34)and on the virtue of Lemma 2.9 and the Borell-Cantelli lemma, the proof will be completed if we show that for any 𝜆>0, _{𝐧∈𝑁𝑟}𝑃(|𝑉_𝐤||𝟐^𝐤−𝟏|>𝜆)<∞,where𝑉_𝐤=‖𝑇^2𝐤_2𝐤−1‖−𝐸‖𝑇^2𝐤_2𝐤−1‖. (2.35)Now, for any 𝜆>0 by Chebyshev inequality and Lemma 2.11, we have_{𝐤∈𝑁𝑟}𝑃(|𝑉_𝐤||𝟐^𝐤−𝟏|>𝜆)≤_{𝐤∈𝑁𝑟}𝐸|𝑉_𝐤|^𝑞𝜆^𝑞|𝟐^𝐤−𝟏|^𝑞≤𝐶_𝑞2^𝑟𝑞𝜆^𝑞_{𝐤∈𝑁𝑟𝟐𝐤−𝟏≤𝐣<𝟐𝐤}𝐸‖𝑌_𝐣‖^𝑞|𝐣|^𝑞≤𝐶_𝑞2^𝑟𝑞𝜆^𝑞_{𝐤∈𝑁𝑟}𝐸‖𝑋_𝐤‖^𝑞|𝐤|^𝑞<∞. (2.36)
(ii) The same arguments and Hölder's inequality_{𝐤∈𝑁𝑟}𝑃(|𝑉_𝐤||𝟐^𝐤−𝟏|𝐶>𝜆)≤^𝑞𝜆_{𝑞𝐤∈𝑁𝑟}1|𝟐^𝐤−𝟏|^𝑞[(_{𝟐𝐤−𝟏≤𝐣<𝟐𝐤}𝐸‖𝑌_𝐣‖²)^𝑞/2+_{𝟐𝐤−𝟏≤𝐣<𝟐𝐤}𝐸‖𝑌_𝐣‖^𝑞]≤𝐶_𝑞𝜆_{𝑞𝐤∈𝑁𝑟}1|𝟐^𝐤−𝟏|^𝑞[|𝟐^𝐤−𝟏|^𝑞/2−1_{𝟐𝐤−𝟏≤𝐣<𝟐𝐤}𝐸‖𝑌_𝐣‖^𝑞+_{𝟐𝐤−𝟏≤𝐣<𝟐𝐤}𝐸‖𝑌_𝐣‖^𝑞]≤2𝐶_𝑞𝜆_𝑞2^{𝑟(𝑞/2+1)}_{𝐤∈𝑁𝑟}𝐸‖𝑋_𝐤‖^𝑞|𝐤|^𝑞/2+1<∞. (2.37)

Acknowledgment

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.