# Strong Laws of Large Numbers for𝔹-Valued  Random Fields

Authors: Zbigniew A. Lagodowski
Journal: Discrete Dynamics in Nature and Society
 (2009)

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## Abstract

We extend to random fields case, the results of Woyczynski, who proved
Brunk's type strong law of large numbers (SLLNs) for𝔹-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.


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## Body

## 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.LetNr, r≥1 be the positive integer r-dimensional lattice points with coordinate
wise partial ordering, ≤. Points in Nr are denoted by m, n or more explicitly n=(n1,n2,…,nr) and 1 stands for the r-tuple (1,…,1). Also for n=(n1,n2,…,nr),
we define |n|=∏i=1rni and (n)={k:k≤n}.
The notation n→∞ means that max1≤i≤rni→∞ or equivalently |n|→∞.Let(Ω,𝔉,P)—be a probability space, (𝔹,∥⋅∥)—a real separable Banach space, {Xk,k∈Nr}—a family of 𝔹-valued random vectors and set (1.1)Sn=∑k≤nXk. If E∥X∥<∞,
then EX stands for the Bochner integral. Let {ak,k∈Nr} be a 𝔹-valued net and a∈𝔹.
We say that an→a strongly as n→∞ if for any ε>0,
there exists Nε∈Nr such that n≰Nε implies ∥an−a∥<ε or shortly for any ε>0, ∥an−a∥≥ε “occurs finitely often” (see Smythe
[1], Fazekas [2]). Furthermore, let {𝔉k,k∈Nr} be an increasing family of sub σ-algebras of 𝔉,
that is,(1.2)⋀k≤n𝔉k⊂𝔉n⊂𝔉.Now, we will introduce
definition of martingale (submartingale) for real-valued random fields, Smythe
[3] (for more information see Merzbach [4]). Through this paper {𝔉k,k∈Nr} satisfies condition CI (conditional
independence)(1.3)E(E(⋅∣𝔉m)∣𝔉n)=E(⋅∣𝔉m⋀n)a.s.,where m⋀n denotes the componentwise minimum of m and n. A{𝔉k,k∈Nr}-adapted, integrable process {Zk,k∈Nr} is called martingale (submartingale)
if(1.4)E(Zn∣𝔉m)=Zm⋀n(≥Zm⋀n)a.s.∀m,n∈Nr.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
submartingales(1.5)λP(maxk≤nSk≥λ)≤ESn+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{Yk,𝔉k,k∈Nr} be a submartingale, {𝔉k,k∈Nr} satisfies (1.3), and let {Ck,k∈Nr} be a nonincreasing array of nonnegative
numbers. Then for λ>0,(1.6)λP(supk≤nCkYk≥λ)≤min1≤s≤r{∑k≤n(Ck−Ck;s;ks+1)EYk+−∑ki,i≠sCk;s;ns∫[⋃ks=1nsBk1,…,kr(s)]cYk;s;ns+dP}≤min1≤s≤r{∑k≤n(Ck−Ck;s;ks+1)EYk+}, where Ck;s;α=Ck1,…,ks−1,α,ks+1,…,kr and Ck=0 if ki>ni for some i=1,2,…,r.Proof.
In the multidimensional martingale case, Theorem1.1 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 Theorem1.1 in martingale case.Remark 1.2.
In the proof of Theorem 2.2 of [11], the authors construct the sets Bk(i) (see the algorithm in [11]) and say “An
explicit expression of the sets Bk(i) in terms of the sets Ak is possible to derive, but such formula is
notationally messy and complicated.” It seems that in the proof, we can
use the sets B˜k(i) constructed as follows (in the case r=2,
for simplicity).
Letn=(n1,n2),
set Zi(ω)=sup1≤j≤n2CijYij(ω),(1.7)I(1)(ω)=inf1≤i≤n1{i:Zi(ω)≥λ}(orn1+1ifnosuchiexists),J(1)(ω)=inf1≤j≤n2{j:CIjYIj(ω)≥λ},     and set (1.8)B˜ij(1)={I(1)(ω)=i,J(1)(ω)=j}.We obtain the sets B˜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, B˜k(1), B˜k(2) are disjoint, 𝔉in2 and 𝔉n1j, respectively, measurable, and(1.9)⋃1≤i≤n1,1≤j≤n2B˜ij(1)=⋃1≤i≤n1,1≤j≤n2B˜ij(2)=[sup1≤i≤n1,1≤j≤n2CijYij≥λ].Such construction gives a simple
formula and is very intuitive.


## 2. The Main Results

We start from
the following generalization of Theorem1.1.Theorem 2.1.
Let{Yk,𝔉k,k∈Nr} be a submartingale, {𝔉k,k∈Nr} satisfies (1.3), and let {Ck,k∈Nr} be a nonincreasing array of nonnegative
numbers. Then for λ>0 and m≤n, (2.1)λP(supk∈[(n)∖(m)]CkYk≥λ)≤min1≤s≤r∑k∈[(n)∖(m)](Ck−Ck;s;ks+1)EYk+, where Ck;s;α=Ck1,…,ks−1,α,ks+1,…,kr, and Ck=0 if ki>ni 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:(2.2)D1={j=(j1,…,jr):m1+1≤j1≤n1,1≤j2≤m2,…,1≤jr≤mr},Di={j=(j1,…jr):1≤j1≤n1,1≤j2≤n2,…,mi+1≤ji≤ni,1≤ji+1≤mi+1,…,1≤jr≤mr},for i=2,3,…,r.
It is easy to
see that⋃i=1rDi=D and Di∩Dj=∅ for i≠j.
Now, let us observe that we can apply Theorem 1.1 to the “cubes” {k∈Nr:l≤k≤n},
where 1≤l<n. Thus we have(2.3)λP(supk∈[(n)∖(m)]CkYk≥λ)=λP(⋃k∈[(n)∖(m)][CkYk≥λ])=λP(⋃i=1r⋃k∈Di[CkYk≥λ])≤λ∑i=1rP(⋃k∈Di[CkYk≥λ])≤λ∑i=1rP(supk∈DiCkYk≥λ)≤∑i=1rmin1≤s≤r∑k∈Di{(Ck−Ck;s;ks+1)EYk+}≤min1≤s≤r∑i=1r∑k∈Di{(Ck−Ck;s;ks+1)EYk+}≤min1≤s≤r∑k∈[(n)∖(m)]{(Ck−Ck;s;ks+1)EYk+}.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,…,ls), m=(m1,…,mt) then Ns+t∋(l,m):=(l1,…,ls,m1,…,mt).Lemma 2.2.
Lets,t≥1 be natural numbers, with s+t=r and {al,l∈Ns},{bm,m∈Nt} families of increasing, positive numbers such
that al→∞,bm→∞ strongly as l and m→∞.
Furthermore {x(l,m),(l,m)∈Nr} be an array of positive numbers such that(2.4)∑(l,m)∈Nrx(l,m)albm<∞.Then(2.5)1aN1∑(l,m)≤(N1,N2)x(l,m)bm→0stronglyasNs∋N1→∞for every N2∈Nt.Proof.
By applying the Martikainen lemma to(2.6)y(l,N2)=∑m≤N2x(l,m)bmwhereN2∈Nt.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
everyn∈Nr and for any family {Xk,k∈Nr} of independent random vectors in 𝔹 with EXk=0,  (2.7)E∥∑k≤nXk∥q≤CE(∑k≤n∥Xn∥p)q/p.Proof.
Forr=1 (Woyczyński [14]) and
since {Xk,k∈Nr} are independent, the lemma is also valid for r>1.Theorem 2.4.
Let1≤p≤2, q>1 and {Xk,k∈Nr} be field of independent, zero-mean, 𝔹-valued random vectors such
that
(2.8)min1≤s≤r∑k∈NrE∥Xk∥pqks|k|pq−q<∞.If 𝔹 is R-type p, then(2.9)Sn|n|→0stronglya.s.asn→∞.Proof.
Let𝔉n=σ(Xk,k≤n),
since Xk are independent, {𝔉k,k∈Nr} satisfies (1.3) and {∥Sk∥pq,𝔉k,k∈Nr} is real, nonnegative submartingale. By
the definition of strong convergence of elements of 𝔹 and “event occurs finitely (infinitely)
often,” it is enough to prove(2.10)limN→∞P(supk≰N∥Sk∥|k|≥λ)=0whereNr∋N=(N,…,N) for any λ>0.  Let us observe, that by Theorem 2.1, Lemma 2.3
and Hölder's inequality for some constants C, we get(2.11)λpqP(supk≰N∥Sk∥|k|≥λ)≤min1≤s≤r∑k≰N(1|k|pq−1(|k|pq;s;ks+1))E∥Sk∥pq≤Cmin1≤s≤r∑k≰N(1|k|pq−1(|k|pq;s;ks+1))E(∑j≤k∥Xj∥p)q≤Cmin1≤s≤r∑k≰N(1|k|pq−1(kpq;s;ks+1))|k|q−1∑j≤kE∥Xj∥pq≤Cmin1≤s≤r∑k≰N1ks|k|pq−q+1∑j≤kE∥Xj∥pq,where (|k|β;s;α):=k1β⋅,…,⋅ks−1βαβks+1β⋅,…,⋅krβ.
Now, it is enough to prove that appropriate multiple
series is finite. Changing the order of summation and comparing to integrals, for
some constantC>0 and for every s,1≤s≤r, we have(2.12)∑k≤N1ks|k|pq−q+1∑j≤kE∥Xj∥pq=∑k≤NE∥Xk∥pq∑k≤j≤N1js|j|pq−q+1≤C∑k≤NE∥Xk∥pq(1Npq−q+1−1kspq−q+1)∏i=1,i≠sr(1Npq−q−1kipq−q).The above expression contains
the following types of sums:(2.13)∑k≤NE∥Xk∥pqNpq−q+1Nl(pq−q)(ki1⋅…⋅kim)pq−q,(2.14)∑k≤NE∥Xk∥pqkspq−q+1Nl(pq−q)(ki1⋅…⋅kim)pq−q,where l,m=0,1,…,r−1,l+m=r−1, and {i1,⋅…⋅,im} is any subset of {1,2,…,r}∖{s}.
Now, by Lemma2.2, (2.13) tends to 0 as N→∞ as well as (2.14) for l≠0. Hence, we have(2.15)∑k∈Nr1ks|k|pq−q+1∑j≤kE∥Xj∥pq≤C∑k∈NrE∥Xk∥pqks|k|pq−q.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).Forr=1, we let obtain the following result.Corollary 2.5.
Let1≤p≤2, q>1 and let {Xk,k∈N} be sequence of independent, zero-mean, 𝔹-valued random vectors such
that (2.16)∑n=1∞E∥Xn∥pqnpq−q+1<∞.If 𝔹 is R-type p, then(2.17)Snn→0a.s.asn→∞.Corollary2.5 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.
Letr=2 and let {Xk,k∈N2} be a field of random vectors fulfilling the
assumptions of Theorem 2.4. For θ∈R+,
we define 2-dimensional sector Nθ2 as follow:(2.18)Nθ2={(k1,k2)∈N2:1≤k2≤θk1}.Assume that {E∥Xn∥pq,n∈N2} are uniformly bounded by constant M and pq−q>1/2. Hence by comparison to integrals, we have(2.19)∑(k1,k2)∈Nθ2E∥X(k1,k2)∥pqks(k1k2)pq−q≤M∑(k1,k2)∈Nθ21ks(k1k2)pq−q<∞,for s=1,2.
Thus, the condition (2.8) of Theorem 2.4 is met and we
have(2.20)∑Nθ2∋k≤nXk|n|→0stronglya.s.asn→∞.
In Theorem2.7, 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.
Let1≤p≤2, q≥1.
The following conditions  are equivalent:
(i)
𝔹 is R-type p.(ii)
For everyλ>0 there exists Cλ such that for any independent, 𝔹-valued, zero-mean random vectors {Xk,k∈Nr},  (2.21)∑k∈Nr|k|−1P(∥Sk∥|k|≥λ)≤Cλ∑k∈NrE∥Xk∥pq|k|pq−q+1.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  (2.22)∑k∈Nr|k|−1P(∥Sk∥|k|≥λ)≤λ−pq∑k∈NrE∥Sk∥pq|k|pq+1≤Cλ−pq∑k∈Nr|k|−pq+q−2∑i≤kE∥Xi∥pq≤Cλ−pq∑k∈NrE∥Xk∥pq∑i≥k|i|−pq+q−2≤Cλ−pq∑k∈NrE∥Xk∥pq|k|pq−q+1.
(ii)⇒(i) Let n=(n1,…,nr) and let {Yn1,n1≥1} be an arbitrary sequence of independent random
vectors in 𝔹,
with EYn1=0 and Tm=∑n1=1mYn1.
Set(2.23)Xn={Yn1for(n1,n2,…,nr)=(n1,1,…,1),0for(n1,n2,…,nr)≠(n1,1,…,1).Then (2.24)∑n1=1∞n1−1P(∥Tn1∥n1≥λ)=∑n∈Nr|n|−1P(∥Sn∥|n|≥λ)≤Cλ∑n∈NrE∥Xn∥pq|n|pq−q+1=Cλ∑n1=1∞E∥Yn1∥pqn1pq−q+1.Thus, (i) follows directly from
Theorem 3.1 of Woyczyński [14].Combining Theorem2.7 with the result of Rosalsky
and Van Thanh [18, Theorem 3.1],  we get the following corollary.Corollary 2.8.
Let1≤p≤2,q≥1, and let 𝔹 be a separable Banach space. If {Xk,k∈Nr} is family of independent, 𝔹-valued, zero-mean random vectors, then the
following conditions are equivalent.
(i)
for everyq≥1 and λ>0, there exists Cλ such that for any vectors {Xk,k∈Nr},  (2.25)∑k∈Nr|k|−1P(∥Sk∥|k|≥λ)≤Cλ∑k∈NrE∥Xk∥pq|k|pq−q+1.(ii)
For every random vectors{Xk,k∈Nr}, the condition(2.26)∑k∈NrE∥Xk∥p|k|p<∞
implies that
the SLLN holds.Before we state
the next theorem, we need more notations and present some useful lemmas. LetN0=(0,1,2,…) and 2n=(2n1,…,2nr),
where 2−1 is defined as 0 and(2.27)fork<ndenoteSkn=∑k<j≤nXj.Lemma (Fazekas [2, Lemma 2.5]).
Let{Xk,k∈Nr} be independent symmetric 𝔹-valued random vectors. Assume that for all λ>0,(2.28)∑k∈N0rP(|∥S2k−12k∥−E∥S2k−12k∥||2k−1|>λ)<∞,E(Sn|n|)→0stronglyasn→∞,then SLLN (2.9) holds.Lemma (Fazekas [2, Lemma 2.3] with ak=|k|).
Let{Xk,k∈Nr} be a field of independent, symmetric, 𝔹-valued random vectors. If(2.29)∥Xk∥≤|k|a.s.foreveryk∈Nr,Sn|n|→0strongly  in  probability,then E∥Sn∥/|n|→0stronglyasn→∞.Lemma 2.11.
Forq≥1, there exists a positive constant Cq such that for any separable Banach space 𝔹 and any finite set {Xk,k≤n} of independent 𝔹-valued random vectors with Xk∈Lq for all k≤n,
the following holds.
For1≤q≤2,  (2.30)E|∥Sn∥−E∥Sn∥|q≤Cq∑k≤nE∥Xk∥q,if q=2, then it is possible to take C2=4.
Forq>2, (2.31)E|∥Sn∥−E∥Sn∥|q≤Cq[(∑k≤nE∥Xk∥2)q/2+∑k≤nE∥Xk∥q].Proof.
Forr=1, the result is due to de Acosta [19, Theorem 2.1].
Since {Xk,k∈Nr} are independent the theorem is true in the
multidimensional case.Theorem 2.12.
Let{Xk,k∈Nr} be a family of independent 𝔹-valued zero-mean, random vectors and assume ∥Sn∥/|n|→0 strongly in probability. Then
(i)
if1≤q≤2, then ∑n∈Nr(E∥Xn∥q/|n|q)<∞ implies SLLN (2.9),(ii)
if2≤q, then ∑n∈Nr(E∥Xn∥q/|n|q/2+1)<∞ implies SLLN (2.9).Theorem2.12
is multiple sum analogue of a strong law of large numbers, Theorem 3.2 of
de Acosta [19].Proof.
(i)
Let us assume that{Xk,k∈Nr} are symmetric (desymmetryzation is standard)
and put(2.32)Yk=XkI(∥Xk∥≤|k|),Tn=∑n∈NrYk.By assumption, it follows that ∑n∈NrP(∥Xk∥≥|k|)<∞ and by the Borell-Cantelli lemma, it is enough
to prove Tn/|n|→0stronglya.s. It follows from assumption that(2.33)Tn|n|→0stronglyinprobability,thus by Lemma 2.10(2.34)E∥Tn∥|n|→0stronglyasn→∞,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,  (2.35)∑n∈NrP(|Vk||2k−1|>λ)<∞,whereVk=∥T2k−12k∥−E∥T2k−12k∥.Now, for any λ>0 by Chebyshev inequality and Lemma 2.11, we
have(2.36)∑k∈NrP(|Vk||2k−1|>λ)≤∑k∈NrE|Vk|qλq|2k−1|q≤Cq2rqλq∑k∈Nr∑2k−1≤j<2kE∥Yj∥q|j|q≤Cq2rqλq∑k∈NrE∥Xk∥q|k|q<∞.
(ii) The same arguments and Hölder's inequality(2.37)∑k∈NrP(|Vk||2k−1|>λ)≤Cqλq∑k∈Nr1|2k−1|q[(∑2k−1≤j<2kE∥Yj∥2)q/2+∑2k−1≤j<2kE∥Yj∥q]≤Cqλq∑k∈Nr1|2k−1|q[|2k−1|q/2−1∑2k−1≤j<2kE∥Yj∥q+∑2k−1≤j<2kE∥Yj∥q]≤2Cqλq2r(q/2+1)∑k∈NrE∥Xk∥q|k|q/2+1<∞.



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