Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2009, Article ID 485412,12 pages doi:10.1155/2009/485412

Research Article

Strong Laws of Large Numbers for B-Valued Random Fields

Zbigniew A. Lagodowski

Department of Mathematics, Faculty of Electrical Engineering and Computer Science, Lublin University of Technology, Nadbystrzycka Street 38A, 20-618 Lublin, Poland

Correspondence should be addressed to Zbigniew A. Lagodowski, z.lagodowski@pollub.pl

Received 30 October 2008; Revised 31 December 2008; Accepted 13 March 2009

Recommended by Stevo Stevic

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.

Copyright © 2009 Zbigniew A. Lagodowski. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 Nr, 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 = (n1f n2,...,nr) and 1 stands for the r-tuple (1,...,1). Also for n = (ni,n2,---,nr), we define |n| = nri=1ni and (n) = {k : k < n}. The notation n ^ to means that maxi<i<rni ^ to or equivalently |n| ^ to.

Let (Q, F,P)—be a probability space, (b, || ■ ||)—a real separable Banach space, {Xk, k e Nr}—a family of b-valued random vectors and set

Sn = ^ Xk.

If E||X|| < go, then EX stands for the Bochner integral. Let {ak, k e Nr} be a b-valued net and a e b. We say that an ^ a strongly as n ^ g if for any e> 0, there exists Ne e Nr such that n / Ne implies ||an - a|| < e or shortly for any e > 0, ||an - a|| > e "occurs finitely often" (see Smythe [1], Fazekas [2]). Furthermore, let {Fk, k e Nr} be an increasing family of sub c-algebras of F, that is,

AFk c Fn c 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 k € Nr} satisfies condition CI (conditional independence)

) = H -IF

where m A n denotes the componentwise minimum of m and n. A(Fk, k e Nr}-adapted, integrable process (Zk, k e Nr} is called martingale (submartingale) if

E(Zn | Fm) = ZmAn( > ZmAn) a.s. Vm,n e Nr. (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 submartingales

Xp(max Sk > A) < ES^ (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 (Yk,Fk, k e Nr} be a submartingale, (Fk, k e Nr} satisfies (1.3), and let (Ck, k e Nr} be a nonincreasing array of nonnegative numbers. Then for 1 > 0,

Xp(sup CkYk > l) < min { v (Ck - Ck;S;fcs+i)EYk+- Y CksnJ YLn dP

k<n / Ks<r[kTn kj/s J [U

< mj<nr{ y<y(Ck - Ck;s;ks + l)EYk+}

s I c" k;s;n

' (1.6)

/ \ c K.s.uc

[I |"s B(s) Ic [Uk. =1 Bk, .....kr J

where Ck;s;« = Ck1/...,ks-1,a,ks+1,...,kr and Ck = 0if ki> n 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

Remark 1.2. In the proof of Theorem 2.2 of [11], the authors construct the sets B^ (see the algorithm in [11]) and say "An explicit expression of the sets B^ 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® constructed as follows (in the case r = 2, for simplicity). Let n = (m,m), set Zi(w) = sup1<j<n2CijYj(w),

I (1)(w) = inf {i: Zi(w) > 1} (or n + 1 if no such i exists),

1<i<n1

J(1)(w)= inf {j : CijYij(w) > 1],

1<j<n2

and set

= {I (1)(w) = i,J (1)(w) = j}. (1.8)

We obtain the sets Bk 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^, B® are disjoint, Fin2 and Fn1j, respectively, measurable, and

U B«= u s? =

sup CijYij > 1

1<i<n1,1<j<n2

1<i<n1,1<j<n2 1<i<n1,1<j<n2

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 {Yk,Fk, k e Nr} be a submartingale, {Fk, k e Nr} satisfies (1.3), and let {Ck, k e Nr} be a nonincreasing array of nonnegative numbers. Then for 1 > 0 and m < n,

Xp( sup CkYk > ^ < min £ (Ck - Ck;S;fcs+i)EYk+, (2.1)

\ke[(n)\(m)] / 1is<r ke[(n)\(m)]

where Cksa = Ckl..,ks-1,a,ks+1,.,kr, and Ck = 0 if ki> m 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 = (}1,...,jr) : m1 + 1 < j < «1, 1 < /2 < m2,...,1 < jr < mr},

Di ={j = (j1, ...jr) : 1 < j1 < «1, 1 < /2 < n2,...,mi + 1 < ji < ni, (2.2)

1 < ji+1 < mi+1,..., 1 < jr < mr},

for i = 2,3,...,r.

It is easy to see that Ur=1 Di = D and Di n Dj = 0 for i = j. Now, let us observe that we can apply Theorem 1.1 to the "cubes" {k e Nr: l < k < n}, where 1 < l < n. Thus we have

1p( sup Ck Yk > l) = 1p( U [CkYk > 1])

\ke[(n)\(m)] / ^ke[(n)\(m)] '

[(n)\(m)]

= 1p(u u [CkYk > 1]

M=1 keDi

< u [CkYk > 1]

i=1 ^keDi

< ^pfsup CkYk > 1) (2.3)

i=1 VkeDi /

<X minEi^k - C№o £Yk+}

i=1 keDi

< imn^SKCk - Ck;S;ks + 0EYk+}

-- i=1 keDi

< m^n £ {(Ck - Ck;S;ks + 1)EYk+ }.

1-S-r ke[(n)\(m)]

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^, m = (m1,...,mt) then Ns+t 3 (l, m) := (h,... ,ls,m1,...,mt).

Lemma 2.2. Let s,t > 1 be natural numbers, with s + t = r and [a\, 1 e Ns}, {bm, m e Nt} families of increasing, positive numbers such that a1 ^ to, bm ^ to strongly as 1 and m ^ to. Furthermore (x(1/m), (1, m) e Nr} be an array of positive numbers such that

EX(1,m)

(1,m) eNr a1bm

X (1,m) aNi (1,m)<(Ni,N2) bm

0 strongly as Ns 3 N1 —> to

for every N2 e Nt.

Proof. By applying the Martikainen lemma to

(1,m) m<N2 bm

where N2 e Nt.

(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 e Nr and for any family {Xk, k e Nr} of independent random vectors in b with EXk = 0,

/ \ q/p

< c^||Xn||M .

\k<n /

Proof. For r = 1 (Woyczyhski [14]) and since {Xk, k e Nr} are independent, the lemma is also valid for r > 1. □

Theorem 2.4. Let 1 < p < 2, q > 1 and {Xk, k e Nr} be field of independent, zero-mean, B-valued random vectors such that

min V E!Xk"P? < (2-8)

liS<r ¿Nr

If B is R-type p, then

I 0 strongly a.s. as n —> to. (2.9)

Proof. Let Fn = o(Xk/ k < n), since Xk are independent, {Fk, k e Nr} satisfies (1.3) and {||Sk||pq, Fk, k e 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

lim p(sup^^ > XJ = 0 where Nr 3 N = (N,...,N) (2.10)

N\k/N |k| /

for any X > 0. Let us observe, that by Theorem 2.1, Lemma 2.3 and Holder's inequality for some constants C, we get

XPqpisup> x\ < min J- - --1--N)£||Sk||pq

\k<N |k| " ) ~1<s<r ¿NX |k|pq (|k|pq; s; ks + 1) J 11 k"

,k<N |k| _ J "Ks<r\k\pq (|k|Pq;s;ks + 1)

- ^2 (^ - (|k|pq;s1 ks + K§

< Cmin y(-1- ---1--N)|k|q-1VE\\Xi\\pq

~ i<s<r kf^V |k|pq (k??; s; ks + 1)/' fk

(2.11)

< Cmin V-1-rV EXr,

- ms<r k/N ks|k|pq-q+1 " ,

where (|k|f; s;a) := kf•,..., •kf-1a^kf+1-,..., -kf.

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

xr^q+r= xEM17 x-j1

k<N ks |kr-q+1 ifk 11 k-N i /s|i|pq-q+1

< Ce^Kt^ -7^) fl

k<N XN ks '1=1,1 /s^ ki '

(2.12)

The above expression contains the following types of sums:

V___(213)

k<NNpq-q+1N1(pq-q)(kh •... • klm)vq-q' v ' '

^^_, (2.14)

kfN k',"-,+1N«pi-q)(ki1 •... • km )'"

where l,m = 0,1,...,r - 1, l + m = r - 1, and {i1, •... •, im} is any subset of {1,2,...,r} \ {s}.

Now, by Lemma 2.2, (2.13) tends to 0 as N ^ to as well as (2.14) for l / 0. Hence, we

1 ^ EX

¿mkp^ § E||Xj||pq < ^kik^ (Z15)

We complete the proof by taking the minimum over s e {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 {Xk, k e N} be sequence of independent, zero-mean, B-valued random vectors such that

E||XJ|pq

v 11 "< to. (2.16)

Z-l "pq-q+l "=1

If B is R-type p, then

0 a.s. as n —> to . (2.17)

Corollary 2.5 for q > 1 is due to Woyczynski [14], which generalized results of Hoffman-Jorgensen, Pisier and Woyczynski [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 {Xk, k e N2} be a field of random vectors fulfilling the assumptions of Theorem 2.4. For 0 e R+, we define 2-dimensional sector N0 as follow:

N2 = {(k1, ki) e N2 : 1 < k2 < 0k1 }. (2.18)

Assume that {E\\Xn\\pq, n e N2} are uniformly bounded by constant M and pq - q > 1/2. Hence by comparison to integrals, we have

E\\X(kh

y ^II-H^Ii - M y -x-< to, (2.19)

¿n ks(kik2)pq-q - (kiMkskk2r-q ' ( )

for s = 1,2.

Thus, the condition (2.8) of Theorem 2.4 is met and we have

en?3 k-nXk

-e—--> 0 strongly a.s. as n to. (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 1 > 0 there exists Cx such that for any independent, B-valued, zero-mean random vectors {Xk, k e Nr},

y | k |WIM > A < Cx y E1M1. (2.21)

kN V i ki " ) ~ XkiNr i ki pq-q+1 1 ;

For r = 1, the theorem is due to Woyczynski [14].

Proof. (i)^(ii) Using Chebyshev inequality ,Lemma 2.3 and Holder's inequality ,we have

kTNr 1 V |k >1 <1 kN |k|pq+1

< CX-pqZ |k|-pq+q-2^E\\Xi\\pq

keNr i<k

(2.22)

< cx-p^E\Xk\P^ |i|-pq+q-2

keNr i>k

< cx-pq y ElXk^. - keNr |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 = ^m=1Yn1. Set

f Yn1 for (m,n2 ,...,nr) = {U1,1,...,1), Xn =\ (2.23)

I 0 for (n1,n2,...,nr) / (n1, 1,..., 1).

M ^ > 0=nsrH-1K l-w >x

^ E\\Xn\\pq < (2.24)

|n|pq-q+1

= Cx'Z

npq-q+1 n1 = 1 n1

Thus, (i) follows directly from Theorem 3.1 of Woyczynski [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 {Xk, k e Nr} is family of independent, B-valued, zero-mean random vectors, then the following conditions are equivalent.

(i) for every q > 1 and 1 > 0, there exists Cx such that for any vectors {Xk, k e Nr},

kSik-p ( x) < (2-)

(ii) For every random vectors {Xk, k e Nr}, the condition

„ E||Xk||p

V 1 < œ (2.26)

|k|p v '

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 2n = (2n1 ,...,2n), where 2-1 is defined as 0 and

for k < n denote Sk = £ Xj. (2.27)

Lemma 2.9 (Fazekas [2, Lemma 2.5]). Let {Xk, k e Nr} be independent symmetric B-valued random vectors. Assume that for all 1 > 0,

2 p ^S2k-" ») <

keN0 V I2 I / (228)

E^Sl^ —> 0 strongly as n —> œ,

then SLLN (2.9) holds.

Lemma 2.10 (Fazekas [2, Lemma 2.3] with ak = |k|). Let {Xk, k e Nr} be afield of independent, symmetric, B-valued random vectors. If

||Xk|| < |k| a.s. for every k e Nr,

Sn (2.29)

|n| —> 0 strongly in probability,

then E||Sn||/|n| ^ 0 strongly as n ^ to.

Lemma 2.11. For q > 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 e Lq for all k < n, the following holds. For 1 < q < 2,

£|||Sn|| - E\\Sn\\\q < Cq £ E||Xk||q, (2.30)

if q = 2, then it is possible to take C2 = 4. For q > 2,

E Sn E Sn

|q < Cq

£E||Xk|

(2.31)

Proof. For r = 1, the result is due to de Acosta [19, Theorem 2.1]. Since {Xk, k e Nr} are independent the theorem is true in the multidimensional case. □

Theorem 2.12. Let {Xk, k e Nr} be a family of independent B-valued zero-mean, random vectors and assume ||Sn||/|n| ^ 0 strongly in probability. Then

(i) if 1 < q < 2, then £neNr(E||Xn||q/|n|q) < to implies SLLN (2.9),

(ii) if 2 < q, then £neNr(E||Xn||q/|n|q/2+1) < to 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 {Xk, k e Nr} are symmetric (desymmetryzation is standard) and put

Yk = XkI(||Xk|| < |k|), Tn = £ Yk. (2.32)

By assumption, it follows that £neNrP(|Xk| > |k|) < to and by the Borell-Cantelli lemma, it is enough to prove Tn/|n| ^ 0 strongly a.s. It follows from assumption that

jnj —> 0 strongly in probability, (2.33)

thus by Lemma 2.10

0 strongly as n to, (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 X> 0,

e p( rsr >X)< where vk = htfk-i y - E y t2ti ii.

neNr \ 2

(2.35)

Now, for any X> 0 by Chebyshev inequality and Lemma 2.11, we have

LH s ») < e E|Vk|'

keNr \ 2

keNX |2

Cq 2rq

Xq ¿-i ¿-J lj\q

X keNr 2k-i<j<2k Ul

Cq 2rq _ E||X]

(2.36)

(ii) The same arguments and Holder's inequality

>) <^E^if EE

keNr \ |2 I / Xq keNr |2 | \2k-i<j<

Xq keNr | Cq e 1

E E||Yj||q

Xq kN |2k-1lq

1<j<2k / 2k-1<j<2k |2k-1 |q/2-1 e E||Yj||q + e E|| Yj ||q

2k-1<j<2k

2k-1<j<2k

< ^2r(q/2+1) e ŒkË < œ.

inq/2+i

keNr l kl

(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.

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