Statistics and Probability Letters 80 (2010) 756–763

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Statistics and Probability Letters

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Strong laws of large numbers for random fields in martingale type pBanach spacesLe Van Dung a,∗, Nguyen Duy Tien ba Faculty of Mathematics, Danang University of Education, 459 Ton Duc Thang, Lien Chieu, Danang, Viet Namb Faculty of Mathematics, Hanoi University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam

a r t i c l e i n f o

Article history:Received 17 June 2009Received in revised form 10 January 2010Accepted 11 January 2010Available online 29 January 2010

MSC:60B1160B1260F1560G42

a b s t r a c t

We extend Marcinkiewicz–Zygmund strong laws for random fields Vn; n ∈ Nd withvalues in martingale type p Banach spaces. Our results are more general and stronger thanthe result of Gut and Stadtmüller (2009) and some other ones.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Let Nd be the positive integer d-dimensional lattice points, where d is a positive integer. Form = (m1, . . . ,md) and n =(n1, . . . , nd) ∈ Nd, notation m ≺ n means that mi ≤ ni, 1 ≤ i ≤ d, |nα| is used for

∏di=1 n

αii , [m, n) =

∏di=1[mi, ni) is a

d-dimensional rectangle and∨di=1(mi < ni) means that there is at least one of m1 < n1, m2 < n2,. . ., md < nd holds. We

write 1 = (1, . . . , 1) ∈ Nd.Consider a random field Vn, n ∈ Nd of random elements defined on a probability space (Ω,F , P) taking values in a

real separable martingale type p (1 ≤ p ≤ 2) Banach spaceXwith norm ‖ · ‖. In the current work, we establish strong lawsof large numbers (SLLN) for |nα|−1maxk≺n ‖Sk‖. This can be done by studying convergence of sums of type∑

n

|n|−1Pmaxk≺n‖Sk‖ > ε|nα| for every ε > 0.

Many authors have investigated the Marcinkiewicz type strong laws of large numbers for random fields Xn, n ∈ Ndof random variables. For example, Fazekas and Tómács (1998) studied strong laws of large numbers |n|−1/rSn (for some0 < r < 1) for pairwise independent random variables, Czerebak-Mrozowicz et al. (2002) studied Marcinkiewicz typestrong laws of large number |n|−1/p(Sn − ESn) (for some 1 < p < 2) for pairwise independent random fields. Recently, Gutand Stadtmüller (2009) studiedMarcinkiewicz–Zygmund laws of large numbers for random fields of i.i.d. random variables.In this paper, we not only extend these results to random fields in martingale type p Banach spaces but also bring moregeneral and stronger ones.Throughout this paper, the symbol C will denote a generic constant (0 < C <∞) which is not necessarily the same one

in each appearance.

∗ Corresponding author.E-mail addresses: lvdunght@gmail.com (L.V. Dung), nduytien2006@yahoo.com (N.D. Tien).

0167-7152/$ – see front matter© 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.spl.2010.01.007

L.V. Dung, N.D. Tien / Statistics and Probability Letters 80 (2010) 756–763 757

2. Preliminaries

Technical definitions relevant to the current work will be discussed in this section.Scalora (1961) introduced the idea of the conditional expectation of a random element in a Banach space. For a random

element V and sub-σ -algebra G of F , the conditional expectation E(V |G) is defined analogously to that in the randomvariable case and enjoys similar properties.A real separable Banach space X is said to be martingale type p (1 ≤ p ≤ 2) if there exists a finite positive constant C

such that for all martingales Sn; n ≥ 1with values inX,

supn≥1E‖Sn‖p ≤ C

∞∑n=1

E‖Sn − Sn−1‖p.

It can be shown using classical methods frommartingale theory that ifX is of martingale type p, then for all 1 ≤ r <∞there exists a finite constant C such that

E supn≥1‖Sn‖r ≤ CE

(∞∑n=1

‖Sn − Sn−1‖p) rp

.

Clearly every real separable Banach space is of martingale type 1 and the real line (the same as any Hilbert space) is ofmartingale type 2. If a real separable Banach space of martingale type p for some 1 < p ≤ 2 then it is of martingale type rfor all r ∈ [1, p).It follows from the Hoffmann-Jørgensen and Pisier (1976) characterization of Rademacher type p Banach spaces that if a

Banach space is of martingale type p, then it is of Rademacher type p. But the notion of martingale type p is only superficiallysimilar to that of Rademacher type p and has a geometric characterization in terms of smoothness. For proofs and moredetails, the reader may refer to Pisier (1975, 1986).To prove the main result we need the following lemma which was proved by Dung et al. (2009) in the case d = 2. If d is

arbitrary positive integer, then the proof is similar and so is omitted.

Lemma 2.1. Let 1 ≤ p ≤ 2. and let Vk, k ≺ n be a collection of |n| random elements in a real separable martingaletype p Banach space with E(Vk|Fk) = 0 for all k ≺ n, where Fk is the σ -field generated by the family of random elementsVl :

∨di=1(li < ki), F1 = ∅,Ω. Then

Emaxk≺n‖Sk‖p ≤ C

∑k≺n

E‖Vk‖p,

where Sk =∑i≺k Vi. In the case of p = 1, the hypothesis that E(Vk|Fk) = 0 for all k ≺ n is superfluous.

Lemma 2.2. Let 1 < p ≤ 2. Let α1, . . . , αd be positive constants satisfying 1/p < minα1 . . . , αd < 1, let q be the number ofintegers s such that αs = minα1 . . . , αd. If E

(‖V‖r(log+ ‖V‖)q−1

)<∞ then we have

(i)∑n

1|nα|

∫∞

|nα |P‖V‖ ≥ tdt <∞,

(ii)∑n

1|nα|p

∫|nα |p

0P‖V‖p ≥ tdt <∞.

Proof. Without loss of generality, we may assume minα1, . . . , αd = α1 = · · · = αq < αq+1 ≤ αd.We first prove (i). We have by Lemma 3 of Stadtmüller and Thalmaier (2009) that

g(j) =∑

1≤n1...nq.nαq+1/α1q+1 ...n

αd/α1d ≤j

1 ∼ Cj(log j)q−1

(q− 1)!as j→∞.

Denote∆g(j) = g(j)− g(j− 1)we get∑n

1|nα|

∫∞

|nα |P‖V‖ ≥ tdt ≤

∞∑k=1

1kα1

∆g(k)∫∞

kα1P‖V‖ ≥ tdt =

∞∑k=1

1kα1

∆g(k)∞∑i=k

∫ (i+1)α1

iα1P‖V‖ ≥ tdt

≤

∞∑k=1

1kα1

∆g(k)∞∑i=k

∫ (i+1)α1

iα1P‖V‖ ≥ iα1dt

≤

∞∑k=1

1kα1

∆g(k)∞∑i=k

iα1−1P‖V‖ ≥ iα1

758 L.V. Dung, N.D. Tien / Statistics and Probability Letters 80 (2010) 756–763

≤

∞∑k=1

1kα1

∆g(k)∞∑i=k

iα1−1∞∑j=i

Pjα1 ≤ ‖V‖ < (j+ 1)α1

≤

∞∑k=1

1kα1

∆g(k)∞∑j=k

Pjα1 ≤ ‖V‖ < (j+ 1)α1j∑i=1

iα1−1

≤ C∞∑k=1

1kα1

∆g(k)∞∑j=k

jα1Pjα1 ≤ ‖V‖ < (j+ 1)α1

≤ C∞∑j=1

jα1Pjα1 ≤ ‖V‖ < (j+ 1)α1j∑k=1

1kα1

∆g(k)

= C∞∑j=1

jα1Pjα1 ≤ ‖V‖ < (j+ 1)α1j∑k=1

1kα1

∆g(k)

= C∞∑j=1

jα1Pjα1 ≤ ‖V‖ < (j+ 1)α1j−1∑k=1

(1kα1−

1(k+ 1)α1

)g(k)

+ C∞∑j=1

Pjα1 ≤ ‖V‖ < (j+ 1)α1g(j)

≤ C∞∑j=1

jα1Pjα1 ≤ ‖V‖ < (j+ 1)α1j−1∑k=1

(1kα1−

1(k+ 1)α1

)k(log k)q−1

+ C∞∑j=1

Pjα1 ≤ ‖V‖ < (j+ 1)α1j(log j)q−1

≤ C∞∑j=1

jα1Pjα1 ≤ ‖V‖ < (j+ 1)α1(log j)q−1j−1∑k=1

(1kα1−

1(k+ 1)α1

)k

+ C∞∑j=1

Pjα1 ≤ ‖V‖ < (j+ 1)α1j(log j)q−1

≤ C∞∑j=1

jα1Pjα1 ≤ ‖V‖ < (j+ 1)α1(log j)q−1j∑k=1

1kα1

+ C∞∑j=1

Pjα1 ≤ ‖V‖ < (j+ 1)α1j(log j)q−1

≤ C∞∑j=1

Pjα1 ≤ ‖V‖ < (j+ 1)α1j(log j)q−1 <∞.

Now we prove (ii).∑n

1|nα|p

∫|nα |p

0P‖V‖p ≥ tdt =

∑n

1|nα|p

∫ 1

0P‖V‖p ≥ tdt +

∑n

1|nα|p

∫|nα |p

1P‖V‖p ≥ tdt

≤ C∑n

1|nα|p

+ C∑n

1|nα|p

∫|nα |p

1P‖V‖p ≥ tdt.

Noting that the first term on the right-hand side is finite, it remains to prove that∑n

1|nα|p

∫|nα |p

1P‖V‖p ≥ tdt.

Denote d(k) =∑n1...nq=k

1, we have by Lemma 3.1 of Gut (2001) that

∞∑j=k

d(j)jpα1

∼ C(log k)q−1

kpα1−1.

L.V. Dung, N.D. Tien / Statistics and Probability Letters 80 (2010) 756–763 759

Hence, we have

∑n

1|nα|p

∫|nα |p

1P‖V‖p ≥ tdt ≤

∞∑k,nq+1,...,nd=1

d(k)1

kpα1 .npαq+1q+1 . . . npαdd

[kα1nαq+1q+1 ...n

αdd ]∑

j=1

E(‖V‖pI(j ≤ ‖V‖ < j+ 1))

(where [x] denotes the greatest integer not exceeding x)

≤ C∞∑

k,nq+1,...,nd=1

d(k)1

kpα1 .npαq+1q+1 . . . npαdd

[kα1nαq+1q+1 ...n

αdd ]∑

j=1

jpP(j ≤ ‖V‖ < j+ 1)

≤ C∞∑

k,nq+1,...,nd=1

d(k)1

kpα1 .npαq+1q+1 . . . npαdd

[kα1nαq+1q+1 ...n

αdd ]∑

j=1

[jp − (j− 1)p]P(‖V‖ ≥ j)

≤ C∞∑

k,nq+1,...,nd=1

d(k)1

kpα1 .npαq+1q+1 . . . npαdd

[kα1nαq+1q+1 ...n

αdd ]∑

j=1

pjp−1P‖V‖ ≥ j

= C∞∑

nq+1,...,nd=1

1

npαq+1q+1 . . . npαdd

∞∑k=1

d(k)kpα1

[kα1nαq+1q+1 ...n

αdd ]∑

j=1

pjp−1P‖V‖ ≥ j

≤ C∞∑

nq+1,...,nd=1

1

npαq+1q+1 . . . npαdd

[nαq+1q+1 ...n

αdd ]∑

j=1

pj(1/α1−1)j(p−1/α1)P‖V‖ ≥ j∞∑k=1

d(k)kpα1

+

∞∑nq+1,...,nd=1

1

npαq+1q+1 . . . npαdd

∞∑i=[n

αq+1q+1 ...n

αdd ]+1

pjp−1P‖V‖ ≥ j∞∑

k=[(j/n

αq+1q+1 ...n

αdd

)1/α1]d(k)kpα1

≤ C∞∑

nq+1,...,nd=1

1

nβq+1q+1 . . . n

βdd

[nαq+1q+1 ...n

αdd ]∑

j=1

pj(1/α1−1)P‖V‖ ≥ j∞∑k=1

d(k)kpα1

+

∞∑nq+1,...,nd=1

1

npαq+1q+1 . . . npαdd

∞∑j=[n

αq+1q+1 ...n

αdd ]+1

pjp−1P‖V‖ ≥ j∞∑

k=[(j/n

αq+1q+1 ...n

αdd

)1/α1]d(k)kpα1

≤ C∞∑

nq+1,...,nd=1

1

nβq+1q+1 . . . n

βdd

∞∑j=1

jr−1(log i)q−1P‖V‖ ≥ j

which is finite if E(‖V‖r log+ ‖V‖)q−1 <∞ and since βl = αl/α1 > 1 for q+ 1 ≤ l ≤ d.

The random field Vn, n ∈ Nd is said to be weakly mean dominated by the random element V if, for some 0 < C <∞,

1|n|

∑k≺n

P‖Vk‖ ≥ x ≤ CP‖V‖ ≥ x

for all n ∈ Nd and x > 0.

3. Main results

With the preliminaries accounted for, the main results may now be established. In the following, we let Vn; n ∈ Ndbe an array of random elements in a real separable Banach space X, Fk is the σ -field generated by the family of randomelements Vl :

∨di=1(li < ki), F1 = ∅,Ω.

The first theorem is a general a.s. convergence one.

Theorem 3.1. Let α1, . . . , αd be positive constants. Let Vn, n ∈ Nd be a random field of random elements. If∑n

1|n|Pmaxk≺n‖Sk‖ > ε|nα| <∞ for every ε > 0, (3.1)

760 L.V. Dung, N.D. Tien / Statistics and Probability Letters 80 (2010) 756–763

then ∑n

Pmaxl≺2n‖Sl‖ > ε′|2nα|

<∞ for every ε′ > 0 (3.2)

and, a fortiori, the SLLN

1|nα|

maxk≺n|Sk| → 0 a.s. as |n| → ∞ (3.3)

obtains.Conversely, (3.2) implies that (3.1) holds.

Proof ((3.1)⇒ (3.2)). Fix ε > 0, denote 2n = (2n1 , . . . , 2nd) and 2nα = (2n1α1 , . . . , 2ndαd). We have the inequalities∑n

Pmaxl≺2n‖Sl‖ > ε|2nα|

≤

∑n

mink∈[2n,2n+1)

Pmaxl≺k‖Sl‖ >

ε

2α1+···,αd|kα|

≤

∑n

∑k∈[2n,2n+1)

1|2n|Pmaxl≺k‖Sl‖ >

ε

2α1+···+αd|kα|

≤

∑n

∑k∈[2n,2n+1)

2d

|k|Pmaxl≺k‖Sl‖ >

ε

2α1+···+αd|kα|

≤ 2d∑n

1|n|Pmaxl≺n‖Sl‖ >

ε

2α1+···+αd|nα|

<∞. (by (3.1))

This implies by the Borel–Cantelli lemma that

1|2nα|

maxk≺2n‖Sk‖ → 0 a.s. as |n| → ∞. (3.4)

Now for k ∈ [2n, 2n+1)we have

1|kα|

maxl≺k‖Sl‖ ≤

1|kα|

maxl≺2n+1

‖Sl‖ ≤1|2nα|

maxl≺2n+1

‖Sl‖ =2α1+···+αd

|2(n+1)α|maxl≺2n+1

‖Sl‖ (3.5)

and so the conclusion (3.3) follows from (3.4) and (3.5).((3.2)⇒ (3.1)). Suppose that (3.2) holds, we easily to prove that for every ε > 0,∑

n

1|n|Pmaxl≺n‖Sl‖ > ε|nα|

≤

∑n

Pmaxl≺2n‖Sl‖ >

ε

2α1+···+αd|2nα|

,

which implies that (3.1) holds. The proof is completed.

The following theorem characterizes the martingale type p Banach spaces.

Theorem 3.2. Let 1 ≤ p ≤ 2 and let X be a separable Banach space. Then the following two statements are equivalent:

(i) The BanachX is of martingale type p.(ii) For every random field Vn; n ∈ Nd inX with E(Vn|Fn) = 0 for all n ∈ Nd and for every α = (α1, . . . , αd) with αi > 0 forall 1 ≤ i ≤ d, the condition∑

n

E‖Vn‖p

|nα|p<∞

implies that, for every ε > 0,∑n

1|n|Pmaxk≺n‖Sk‖ > ε|nα| <∞

and, a fortiori, the SLLN1|nα|

maxk≺n‖Sk‖ → 0 a.s. as |n| → ∞

obtains.

L.V. Dung, N.D. Tien / Statistics and Probability Letters 80 (2010) 756–763 761

Proof. In order to prove [(i)⇒ (ii)]we show that∑n

Pmaxk≺2n‖Sk‖ > ε|2nα| <∞ for every ε > 0.

Applying Markov’s inequality and Lemma 2.2 we have that∑n

Pmaxk≺2n‖Sk‖ > ε|2nα| ≤

∑n

1εp|2nα|p

E(maxk≺2n‖Sk‖p)

≤ C∑n

1|2nα|p

∑k≺2nE‖Vk‖p ≤ C

∑k

E‖Vk‖p

|kα|p<∞.

Nowweprove [(ii)⇒ (i)]. Assume that (ii) holds. Let Wn1 ,Gn1; n1 ≥ 1be an arbitrary sequence ofmartingale differenceinX such that

∞∑n1=1

E‖Wn1‖p

np1<∞

Set

Vn1,...,nd = Wn1 if n2 = · · · = nd = 1 otherwise Vn1,...,nd = 0.

Then Vn1,...,nd is the random field inX satisfies E(Vn1,...,nd |Fn1,...,nd) = 0 for all (n1, . . . , nd) ∈ Nd, and∞∑

n1,...,nd=1

E‖Vn1,...,nd‖p

(n1 . . . nd)p=

∞∑n1=1

E‖Wn1‖p

np1<∞.

By (ii),

1n1 . . . nd

∑i1≤n1id≤nd

Vi1,...,id → 0 a.s. as (n1 . . . nd)→∞.

Taking n2 = · · · = nd = 1 and letting n1 →∞we obtain

1n1

n1∑j=1

Wj → 0 a.s. as n1 →∞.

Then by Theorem 2.2 of Hoffmann-Jørgensen and Pisier (1976),X is of martingale type p.

In the next two theorems, we obtain the Marcinkiewicz–Zygmund type laws of large numbers for random fields ofrandom elements.

Theorem 3.3. Let X be a martingale type p Banach space with 1 < p ≤ 2. Let α1, . . . , αd be positive constants satisfying1/p < minα1 . . . , αd < 1, let q be the number of integers s such that αs = minα1 . . . , αd and let Vn, n ∈ Nd be arandom field satisfying E(Vn|Fn) = 0 for all n ∈ Nd. Suppose that Vn, n ∈ Nd is weakly mean dominated by V such thatE(‖V‖r(log+ ‖V‖)q−1

)<∞ with r = 1

minα1...,αd. Then∑

n

1|n|Pmaxk≺n‖Sk‖ > ε|nα| <∞ (3.6)

and, a fortiori, the SLLN

1|nα|

maxk≺n‖Sk‖ → 0 a.s. as |n| → ∞ (3.7)

obtains.

Proof. For each n ∈ Nd, setV ′k = VkI(‖Vk‖ ≤ |n

α|), V ′′k = VkI(‖Vk‖ > |n

α|),

Y ′k = V′

k − E(V′

k|Fk), Y′′

k = V′′

k − E(V′′

k |Fk),S ′n =

∑k≺n Y

′

k, S′′n =

∑k≺n Y

′′

k .Since E(Vk|Fk) = 0, it follows that Vk = Y ′k + Y

′′

k . Moreover, if G′

k and G′′k are the σ -fields generated by the family ofrandom elements Y ′l :

∨di=1(li < ki) and Y

′′

l :∨di=1(li < ki), respectively, then G′k ⊂ Fk and G′′k ⊂ Fk for all k ≺ n, which

imply that E(Y ′k|G′

k) = E(Y′′

k |G′′

k ) = 0 for all k ≺ n.

762 L.V. Dung, N.D. Tien / Statistics and Probability Letters 80 (2010) 756–763

We now begin the proof. For every ε > 0,∑n

1|n|Pmaxk≺n‖Sk‖ > 2ε|nα| ≤

∑n

1|n|Pmaxk≺n‖S ′k‖ > ε|nα| +

∑n

1|n|Pmaxk≺n‖S ′′k ‖ > ε|nα|. (3.8)

First, we show that∑n

1|n|Pmaxk≺n‖S ′′k ‖ > ε|nα| <∞.

Applying Markov’s inequality and Lemma 2.2, we obtain∑n

1|n|Pmaxk≺n‖S ′′k ‖ > ε|nα| ≤

∑n

1ε|n||n|α

E(maxk≺n‖S ′′k ‖) ≤ C

∑n

1|nα|

1|n|

∑k≺n

E‖Y ′′k ‖

≤ C∑n

1|nα|

1|n|

∑k≺n

E‖V ′′k ‖ = C∑n

1|nα|

1|n|

∑k≺n

∫∞

0P‖V ′′k ‖ ≥ tdt

≤ C∑n

1|nα|

1|n|

∑k≺n

∫|nα |

0P‖Vk‖ ≥ |nα|dt + C

∑n

1|nα|

1|n|

∑k≺n

∫∞

|nα |P‖Vk‖ ≥ tdt

= C∑n

1|n|

∑k≺n

P‖Vk‖ ≥ |nα| + C∑n

1|nα|

∫∞

|nα |

1|n|

∑k≺n

P‖Vk‖ ≥ tdt

≤ C∑n

P‖V‖ ≥ |nα| + C∑n

1|nα|

∫∞

|nα |P‖V‖ ≥ tdt

≤ C∑n

1|nα|

∫∞

|nα |P‖V‖ ≥ tdt <∞ (by Lemma 2.2).

By (3.8), in order to complete the proof, we next show that∑n

1|n|Pmaxk≺n‖S ′k‖ > ε|nα| <∞.

Again applying Markov’s inequality, we find that∑n

1|n|Pmaxk≺n‖S ′k‖ > ε|nα| ≤

∑n

1|n|

1ε|nα|p

E(maxk≺n‖S ′k‖ > ε|nα|)p

=

∑n

1|n|

1ε|nα|p

E(maxk≺n‖S ′k‖ > ε|nα|p) ≤ C

∑n

1|nα|p

1|n|

∑k≺n

E‖Y ′k‖p

≤ C∑n

1|nα|p

1|n|

∑k≺n

E‖V ′k‖p= C

∑n

1|nα|p

1|n|

∑k≺n

∫∞

0P‖V ′k‖

p≥ tdt

≤ C∑n

1|nα|p

1|n|

∑k≺n

∫|nα |p

0P‖Vk‖p ≥ tdt

= C∑n

1|nα|p

∫|nα |p

0

1|n|

∑k≺n

P‖Vk‖p ≥ tdt

≤ C∑n

1|nα|p

∫|nα |p

0P‖V‖p ≥ tdt <∞ (by Lemma 2.2).

The proof is completed.

Remark. Note that in the case of q < d, positive constants α1, . . . , αd are not upper bounded by 1, which is weaker thancondition (2.1) of Theorem 2.1 of Gut and Stadtmüller (2009).

Theorem 3.4. Let α1, . . . , αd be positive constants satisfying minα1 . . . , αd > 1, let q be the number of integers s such thatαs = minα1 . . . , αd. Suppose that Vn, n ∈ Nd is weakly mean dominated by V such that E

(‖V‖(log+ ‖V‖)q−1

)<∞. Then

(3.3) holds and then, the SLLN (3.4) obtains.

Proof. The proof is similar to that of Theorem 3.2 with p = 1 and we use T ′n =∑k≺n V

′

k and T′′n =

∑k≺n V

′′

k are instead ofS ′n and S

′′n , respectively.

L.V. Dung, N.D. Tien / Statistics and Probability Letters 80 (2010) 756–763 763

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