on the laws of large numbers for double arrays of independent random elements in banach spaces

Acta Mathematica Sinica, English Series

Aug., 2014, Vol. 30, No. 8, pp. 1353–1364

Published online: July 15, 2014

DOI: 10.1007/s10114-014-3507-7

Http://www.ActaMath.com

Acta Mathematica Sinica, English Series© Springer-Verlag Berlin Heidelberg & The Editorial Office of AMS 2014

On the Laws of Large Numbers for Double Arrays of Independent

Random Elements in Banach Spaces

Andrew ROSALSKYDepartment of Statistics, University of Florida, Gainesville, Florida 32611-8545, USA

E-mail : [email protected]

Le Van THANH Nguyen Thi THUYDepartment of Mathematics, Vinh University, Nghe An 42118, Vietnam

E-mail : [email protected] [email protected]

Abstract For a double array of independent random elements {Vmn, m ≥ 1, n ≥ 1} in a real separable

Banach space, conditions are provided under which the weak and strong laws of large numbers for the

double sums∑m

i=1

∑nj=1 Vij , m ≥ 1, n ≥ 1 are equivalent. Both the identically distributed and the

nonidentically distributed cases are treated. In the main theorems, no assumptions are made concerning

the geometry of the underlying Banach space. These theorems are applied to obtain Kolmogorov,

Brunk–Chung, and Marcinkiewicz–Zygmund type strong laws of large numbers for double sums in

Rademacher type p (1 ≤ p ≤ 2) Banach spaces.

Keywords Real separable Banach space, double array of independent random elements, strong and

weak laws of large numbers, almost sure convergence, convergence in probability, Rademacher type p

Banach space

MR(2010) Subject Classification 60F05, 60F15, 60B11, 60B12

1 Introduction

Throughout this paper, we consider a double array {Vmn, m ≥ 1, n ≥ 1} of independent randomelements defined on a probability space (Ω,F , P ) and taking values in a real separable Banachspace X with norm ‖ · ‖. We provide conditions under which the strong law of large numbers(SLLN) and the weak law of large numbers (WLLN) for the double sums

∑mi=1

∑nj=1 Vij are

equivalent. Such double sums differ substantially from the partial sums∑n

i=1 Vi, n ≥ 1 of asequence of independent random elements {Vn, n ≥ 1} because of the partial (in lieu of linear)ordering of the index set {(i, j), i ≥ 1, j ≥ 1}. We treat both the independent and identicallydistributed (i.i.d.) and the independent but nonidentically distributed cases. In the main results(Theorems 3.1 and 3.7), no assumptions are made concerning the geometry of the underlyingBanach space. We then apply the main results to obtain Kolmogorov, Brunk–Chung, andMarcinkiewicz–Zygmund type SLLNs for double sums in Rademacher type p (1 ≤ p ≤ 2)Banach spaces.

Received September 6, 2013, accepted December 30, 2013

The second author is supported by the Vietnam Institute for Advanced Study in Mathematics (VIASM)

and the Vietnam National Foundation for Sciences and Technology Development NAFOSTED (Grant No.

101.01.2012.13); the third author is supported by NAFOSTED (Grant No. 101.03.2012.17)

1354 Rosalsky A., et al.

While in the current work attention is restricted to considering double sums, the resultscan of course be extended by the same method to multiple sums over lattice points of anydimension.

The reader may refer to Rosalsky and Thanh [14] for a brief discussion of a historical natureconcerning double sums and on their importance in the field of statistical physics. For the case ofi.i.d. real-valued random variables, a major surrey article concerning double sums was preparedby Pyke [13]. In Pyke [13], he discussed fluctuation theory, the limiting Brownian sheet, theSLLN, and various other limit theorems. Currently, Professor Oleg I. Klesov (National TechnicalUniversity of Ukraine) is preparing a comprehensive book on multiple sums of independentrandom variables (see [9]).

The plan of the paper is as follows. Notation, technical definitions, and five known lemmaswhich are used in proving the main results are consolidated into Section 2. In Section 3, weestablish the main results after first proving three new lemmas. The applications of the mainresults are presented in Section 4. Section 5 contains an example pertaining to Theorems 3.1and 4.1.

2 Preliminaries

In this section, notation, technical definitions, and lemmas which are needed in connection withthe main results will be presented.

For a, b ∈ R, min{a, b} and max{a, b} will be denoted, respectively, by a ∧ b and a ∨ b.Throughout this paper, the symbol C will denote a generic constant (0 < C < ∞) which is notnecessarily the same one in each appearance.

The expected value or mean of an X -valued random element V , denoted by EV , is definedto be the Pettis integral provided it exists. If E‖V ‖ < ∞, then (see, e.g., Taylor [18, p. 40]) V

has an expected value. But the expected value can exist when E‖V ‖ = ∞. For an example,see Taylor [18, p. 41].

Hoffmann-Jørgensen and Pisier [8] proved for 1 ≤ p ≤ 2 that a real separable Banach spaceis of Rademacher type p if and only if there exists a constant C such that

E

∥∥∥∥

n∑

j=1

Vj

∥∥∥∥

p

≤ C

n∑

j=1

E‖Vj‖p (2.1)

for every finite collection {V1, V2, . . . , Vn} of independent mean 0 random elements.For the double array of random elements {Vmn, m ≥ 1, n ≥ 1}, we write

S(m, n) = Smn =m∑

i=1

n∑

j=1

Vij , m ≥ 1, n ≥ 1.

For sums of independent random elements, the first lemma provides in (2.2) and (2.3) aMarcinkiewicz–Zygmund type inequality and a Rosenthal type inequality, respectively. Lemma2.1 is due to de Acosta [1, Theorem 2.1].

Lemma 2.1 Let {Vj , 1 ≤ j ≤ n} be a collection of n independent random elements. Then forevery p ≥ 1, there is a positive constant Cp < ∞ depending only on p such that

E

∣∣∣∣

∥∥∥∥

n∑

j=1

Vj

∥∥∥∥ − E

∥∥∥∥

n∑

j=1

Vj

∥∥∥∥

∣∣∣∣

p

≤ Cp

n∑

j=1

E‖Vj‖p for 1 ≤ p ≤ 2, (2.2)

Laws of Large Numbers for Double Arrays 1355

and

E

∣∣∣∣

∥∥∥∥

n∑

j=1

Vj

∥∥∥∥ − E

∥∥∥∥

n∑

j=1

Vj

∥∥∥∥

∣∣∣∣

p

≤ Cp

(( n∑

j=1

E‖Vj‖2

) p2

+n∑

j=1

E‖Vj‖p

)

for p > 2. (2.3)

The following lemma is due to Hoffmann-Jørgensen [7]; see the proof of Theorem 3.1 in [7].

Lemma 2.2 Let {Vj , 1 ≤ j ≤ n} be a collection of n independent symmetric random elementsin a real separable Banach space. Then for all t > 0, s > 0,

P

(∥∥∥∥

n∑

j=1

Vj

∥∥∥∥ > 2t + s

)

≤ 4P 2

(∥∥∥∥

n∑

j=1

Vj

∥∥∥∥ > t

)

+ P(

max1≤j≤n

‖Vj‖ > s).

The next lemma is Levy’s inequality for double arrays of independent symmetric randomelements in Banach spaces. It is due to Etemadi [3, Corollary 1.2]. We note that Etemadi [3]established the result for d-dimensional arrays where d is arbitrary positive integer.

Lemma 2.3 Let {Vij , 1 ≤ i ≤ m, 1 ≤ j ≤ n} be a collection of mn independent symmetricrandom elements in a real separable Banach space. Then there exists a constant C such that,for all t > 0,

P(

maxk≤m,l≤n

‖Skl‖ > t)≤ CP

(

‖Smn‖ >t

C

)

.

The following result is a double sum analogue of the Toeplitz lemma (see, e.g., Loeve [10,p. 250]) and is due to Stadtmuller and Thanh [17, Lemma 2.2].

Lemma 2.4 Let {amnij , 1 ≤ i ≤ m + 1, 1 ≤ j ≤ n + 1, m ≥ 1, n ≥ 1} be an array of positiveconstants such that

supm≥1,n≥1

m+1∑

i=1

n+1∑

j=1

amnij ≤ C and limm∨n→∞ amnij = 0 for every fixed i, j.

If {xmn, m ≥ 1, n ≥ 1} is a double array of constants with

limm∨n→∞ xmn = 0,

then

limm∨n→∞

m+1∑

i=1

n+1∑

j=1

amnijxij = 0.

The last lemma in this section has an easy proof; see Rosalsky and Thanh [15, Lemma 2.1].

Lemma 2.5 Let {Vmn, m ≥ 1, n ≥ 1} be a double array of random elements in a real separableBanach space and let p > 0. If

∞∑

m=1

∞∑

n=1

E‖Vmn‖p < ∞,

thenVmn → 0 almost surely (a.s.) and in Lp as m ∨ n → ∞.

Finally, we note that the Borel–Cantelli lemma (both the convergence and divergence halves)carries over to an array of events {Amn, m ≥ 1, n ≥ 1} since the sets {(m, n) : m ≥ 1, n ≥ 1}and {k : k ≥ 1} are in one-to-one correspondence with each other. Of course, for the divergencehalf, it is assumed that the array {Amn, m ≥ 1, n ≥ 1} is comprised of independent events.


3 Main Results

With the preliminaries accounted for, the first main result may be established. Theorem 3.1considers the independent but nonidentically distributed case while Theorem 3.7 considersthe i.i.d. case. In these theorems, no assumptions are made concerning the geometry of theunderlying Banach space. In Theorem 3.1, the condition (3.1) is a Kolmogorov type conditionfor the SLLN for double arrays whereas the condition (3.4) is a Brunk–Chung type conditionfor the SLLN for double arrays.

Theorem 3.1 Let α > 0, β > 0 and let {Vmn, m ≥ 1, n ≥ 1} be a double array of independentrandom elements in a real separable Banach space.

(i) Assume that∞∑

m=1

∞∑

n=1

E‖Vmn‖p

mαpnβp< ∞ for some 1 ≤ p ≤ 2. (3.1)

ThenSmn

mαnβ

P→ 0 as m ∨ n → ∞ (3.2)

if and only ifSmn

mαnβ→ 0 a.s. as m ∨ n → ∞. (3.3)

(ii) Assume that∞∑

m=1

∞∑

n=1

E‖Vmn‖2p

m2αp+1−pn2βp+1−p< ∞ for some p > 1. (3.4)

Then (3.2) and (3.3) are equivalent.

The proof of Theorem 3.1 has several steps, so we will break it up into three lemmas. Someof the lemmas may be of independent interest. The first lemma ensures that in Theorem 3.1,it suffices to assume that the array {Vmn, m ≥ 1, n ≥ 1} is comprised of symmetric randomelements. Lemma 3.2 is a double sum analogue (with more general norming constants) ofEtemadi [2, Lemma 1].

Lemma 3.2 Let α > 0, β > 0 and let V = {Vmn, m ≥ 1, n ≥ 1} and V ′ = {V ′mn, m ≥ 1, n ≥

1} be two double arrays of independent random elements in a real separable Banach space suchthat V and V ′ are independent copies of each other. Then

Smn

mαnβ→ 0 a.s. as m ∨ n → ∞ (3.5)

if and only if∑m

i=1

∑nj=1(Vij − V ′

ij)mαnβ

→ 0 a.s. as m ∨ n → ∞ andSmn

mαnβ

P→ 0 as m ∨ n → ∞. (3.6)

Proof The implication (3.5)⇒(3.6) is obvious. To prove the implication (3.6)⇒(3.5), set

S′mn =

m∑

i=1

n∑

j=1

V ′ij , m ≥ 1, n ≥ 1.

Then for all m ≥ 1, n ≥ 1, ‖Smn‖mαnβ and ‖S′

mn‖mαnβ are i.i.d. real-valued random variables. Let μmn

denote a median of ‖Smn‖mαnβ , m ≥ 1, n ≥ 1. By the second half of (3.6),

μmn → 0 as m ∨ n → ∞. (3.7)


By the strong symmetrization inequality (see, e.g., Gut [6, p. 134]), we have for all ε > 0,

P

(

supk≤m∨n≤l

∣∣∣∣

∥∥∥∥

Smn

mαnβ

∥∥∥∥ − μmn

∣∣∣∣ > ε

)

≤ 2P

(

supk≤m∨n≤l

∣∣∣∣

∥∥∥∥

Smn

mαnβ

∥∥∥∥ −

∥∥∥∥

S′mn

mαnβ

∥∥∥∥

∣∣∣∣ > ε

)

≤ 2P

(

supm∨n≥k

∥∥∥∥

Smn − S′mn

mαnβ

∥∥∥∥ > ε

)

→ 0 as k → ∞ (by the first half of (3.6)). (3.8)

Letting l → ∞ in (3.8), we have P(supm∨n≥k

∣∣‖ Smn

mαnβ ‖ − μmn

∣∣ > ε

) → 0 as k → ∞. Thismeans that

Smn

mαnβ− μmn → 0 a.s. as m ∨ n → ∞. (3.9)

By combining (3.7) and (3.9), we obtain (3.5). �The second lemma shows that if ‖Vmn‖ ≤ mαnβ a.s., m ≥ 1, n ≥ 1, then Smn

mαnβ obeying theWLLN as m∨ n → ∞ is indeed equivalent to its convergence in Lp to 0 as m∨ n → ∞ for anyp > 0.

Lemma 3.3 Let α > 0, β > 0 and let {Vmn, m ≥ 1, n ≥ 1} be a double array of independentsymmetric random elements in a real separable Banach space such that ‖Vmn‖ ≤ mαnβ a.s. forall m ≥ 1, n ≥ 1. If

Smn

mαnβ

P→ 0 as m ∨ n → ∞, (3.10)

then for all p > 0,Smn

mαnβ→ 0 in Lp as m ∨ n → ∞. (3.11)

Proof Let p > 0 and let ε > 0 be arbitrary. Let Kmn = max1≤i≤m,1≤j≤n ‖Vij‖, m ≥ 1, n ≥ 1.Since ‖Vmn‖ ≤ mαnβ a.s. for all m ≥ 1, n ≥ 1,

Kmn ≤ mαnβ a.s., m ≥ 1, n ≥ 1. (3.12)

By (3.10), there exists a positive integer N such that whenever m ∨ n ≥ N ,

P (‖Smn‖ ≥ mαnβε) ≤ 18 × 3p

. (3.13)

Now for all A > 0,∫ A

0

tp−1P (‖Smn‖ > mαnβt)dt = 3p

∫ A3

0

tpP (‖Smn‖ > 3mαnβt)dt

≤ 3p

[

4∫ A

3

0

tp−1P 2(‖Smn‖ > mαnβt)dt

+∫ A

3

0

tp−1P (Kmn > mαnβt)dt

]

(by Lemma 2.2)

≤ 4(3ε)p

p+

12

∫ A3

ε

tp−1P (‖Smn‖ > mαnβt)dt

+ 3p

∫ A3

0

tp−1P (Kmn > mαnβt)dt (by (3.13))


≤ 4(3ε)p

p+

12

∫ A

0


+ 3p

∫ 1

0

tp−1P (Kmn > mαnβt)dt (by (3.12)). (3.14)

It follows from (3.14) that for all A > 0,∫ A

0

tp−1P (‖Smn‖ > mαnβt)dt ≤ 8(3ε)p

p+ 2 × 3p

∫ 1

0

tp−1P (Kmn > mαnβt)dt

and hence

E

∥∥∥∥

Smn

mαnβ

∥∥∥∥

p

= p

∫ ∞

0


= p limA→∞

∫ A

0


≤ 8(3ε)p + (2p)3p

∫ 1

0

tp−1P (Kmn > mαnβt)dt. (3.15)

Note that for all m ≥ 1, n ≥ 1, Kmn ≤ 4 maxk≤m,l≤n ‖Skl‖ and so by Lemma 2.3 and (3.10) wehave for t > 0,

P (Kmn > mαnβt) ≤ P

(

maxk≤m,l≤n

‖Skl‖ >mαnβt

4

)

≤ CP

(

‖Smn‖ >mαnβt

4C

)

→ 0 as m ∨ n → ∞. (3.16)

Hence, by the Lebesgue dominated convergence theorem, (3.16) implies that∫ 1

0

tp−1P (Kmn > mαnβt)dt → 0 as m ∨ n → ∞. (3.17)

The conclusion (3.11) follows from (3.15), (3.17) and the arbitrariness of ε > 0. �We use the Levy type inequality for double arrays of independent symmetric random ele-

ments (Lemma 2.3) as a key tool to prove the following lemma. This lemma is a double sumversion of de Acosta [1, Lemma 3.2].

Lemma 3.4 Let α > 0, β > 0 and let {Vmn, m ≥ 1, n ≥ 1} be a double array of independentsymmetric random elements in a real separable Banach space. Then

Smn

mαnβ→ 0 a.s. as m ∨ n → ∞ (3.18)

if and only if∑2m−1

i=2m−1

∑2n−1j=2n−1 Vij

2mα2nβ→ 0 a.s. as m ∨ n → ∞. (3.19)

Proof Let

Tmn =2m−1∑

i=2m−1

2n−1∑

j=2n−1

Vij , m ≥ 1, n ≥ 1.

The implication (3.18) ⇒ (3.19) is immediate since for all m ≥ 2, n ≥ 2,

Tmn = S(2m − 1, 2n − 1)− S(2m − 1, 2n−1 − 1)− S(2m−1 − 1, 2n − 1) + S(2m−1 − 1, 2n−1 − 1).


Next, we assume that (3.19) holds. Since the array {Tmn, m ≥ 1, n ≥ 1} is comprised ofindependent random elements, by the Borel–Cantelli lemma,

∞∑

m=1

∞∑

n=1

P (‖Tmn‖ > 2mα2nβε) < ∞ for all ε > 0. (3.20)

Let

Mrs = max2r−1≤u≤2r−1,2s−1≤v≤2s−1

∥∥∥∥

u∑

i=2r−1

v∑

j=2s−1

Vij

∥∥∥∥, r ≥ 1, s ≥ 1.

By Lemma 2.3 and (3.20), we have∞∑

r=1

∞∑

s=1

P (Mrs > 2rα2sβε) ≤ C∞∑

r=1

∞∑

s=1

P

(

‖Trs‖ >2rα2sβε

C

)

< ∞ for all ε > 0.

This ensures thatMrs

2rα2sβ→ 0 a.s. as r ∨ s → ∞. (3.21)

For m ≥ 1, n ≥ 1, let k ≥ 0, l ≥ 0 be such that

2k ≤ m ≤ 2k+1 − 1 and 2l ≤ n ≤ 2l+1 − 1.

Then for m ≥ 1, n ≥ 1,

‖Smn‖ ≤k+1∑

r=1

l+1∑

s=1

Mrs

and so ∥∥∥∥

Smn

mαnβ

∥∥∥∥ ≤

k+1∑

r=1

l+1∑

s=1

2rα2sβ

2kα2lβ.

Mrs

2rα2sβ. (3.22)

Note that

supk≥1,l≥1

k+1∑

r=1

l+1∑

s=1

2rα2sβ

2kα2lβ< ∞ and lim

k∨l→∞2rα2sβ

2kα2lβ= 0 for every fixed r, s. (3.23)

Hence, from (3.21) and (3.23), we get by applying Lemma 2.4 that

k+1∑

r=1

l+1∑

s=1

2rα2sβ

2kα2lβ.

Mrs

2rα2sβ→ 0 a.s. as k ∨ l → ∞. (3.24)

The conclusion (3.18) follows from (3.22) and (3.24). �Proof of Theorem 3.1(i) Assume that (3.2) holds. By Lemma 3.2, it is enough to prove thetheorem assuming the {Vmn, m ≥ 1, n ≥ 1} are symmetric. Set

Wmn = VmnI(‖Vmn‖ ≤ mαnβ), m ≥ 1, n ≥ 1.

By Markov’s inequality and (3.1),∞∑

n=1

∞∑

m=1

P(‖Vmn‖ > mαnβ

) ≤∞∑

n=1

∞∑

m=1

E‖Vmn‖p

mαpnβp< ∞. (3.25)

Also by (3.1),∞∑

n=1

∞∑

m=1

E‖Wmn‖p

mαpnβp< ∞. (3.26)


By (3.25) and the Borel–Cantelli lemma, it suffices to prove∑m

i=1

∑nj=1 Wij

mαnβ→ 0 a.s. as m ∨ n → ∞. (3.27)

Using (3.25) and the Borel–Cantelli lemma again, it follows from (3.2) that∑m

i=1

∑nj=1 Wij

mαnβ

P→ 0 as m ∨ n → ∞.

Thus, Lemma 3.3 ensures that∑m

i=1

∑nj=1 Wij

mαnβ→ 0 in L1 as m ∨ n → ∞

and so∑2m−1

i=2m−1

∑2n−1j=2n−1 Wij

2mα2nβ=

∑2m−1i=1

∑2n−1j=1 Wij

2mα2nβ−

∑2m−1i=1

∑2n−1−1j=1 Wij

2β2mα2(n−1)β

−∑2m−1−1

i=1

∑2n−1j=1 Wij

2α2(m−1)α2nβ+

∑2m−1−1i=1

∑2n−1−1j=1 Wij

2α2β2(m−1)α2(n−1)β

→ 0 in L1 as m ∨ n → ∞. (3.28)

Now if we can show that

‖∑2m−1i=2m−1

∑2n−1j=2n−1 Wij‖ − E‖∑2m−1

i=2m−1

∑2n−1j=2n−1 Wij‖

2mα2nβ→ 0 a.s. as m ∨ n → ∞, (3.29)

then it follows from (3.28) that∑2m−1

i=2m−1

∑2n−1j=2n−1 Wij

2mα2nβ→ 0 a.s. as m ∨ n → ∞

which yields (3.27) via Lemma 3.4. To prove (3.29), note that

∞∑

m=1

∞∑

n=1

E∣∣‖∑2m−1

i=2m−1

∑2n−1j=2n−1 Wij‖ − E‖∑2m−1

i=2m−1

∑2n−1j=2n−1 Wij‖

∣∣p

2mαp2nβp

≤ C

∞∑

m=1

∞∑

n=1

∑2m−1i=2m−1

∑2n−1j=2n−1 E‖Wij‖p

2mαp2nβp(by (2.2) of Lemma 2.1)

≤ C

∞∑

k=1

∞∑

l=1

E‖Wkl‖p

kαplβp< ∞ (by (3.26)). (3.30)

By Lemma 2.5, (3.29) follows from (3.30). The proof of Theorem 3.1(i) is complete. The proofof Theorem 3.1(ii) is similar and we omit the details but we point out that (2.3) of Lemma 2.1is used instead of (2.2). �

Corollary 3.5 Let α > 0, β > 0 and let {Vmn, m ≥ 1, n ≥ 1} be a double array of independentrandom elements in a real separable Banach space.

(i) Assume that for some 1 ≤ p ≤ 2, some ε > 0, and all m ≥ 1, n ≥ 1,

E‖Vmn‖p ≤ Cmαp−1nβp−1

((log(m + 1))(log(n + 1)))1+ε. (3.31)



(ii) Assume that for some p > 1, some ε > 0, and all m ≥ 1, n ≥ 1,

E‖Vmn‖2p ≤ Cmp(2α−1)np(2β−1)

((log(m + 1))(log(n + 1)))1+ε. (3.32)


Proof (i) Note that by (3.31),

∞∑

m=1

∞∑

n=1

E‖Vmn‖p

mαpnβp≤ C

∞∑

m=1

∞∑

n=1

1m(log(m + 1))1+εn(log(n + 1))1+ε

< ∞

and the result follows from Theorem 3.1(i). The proof of part (ii) is similar. �

Remark 3.6 Suppose that supm≥1,n≥1 E‖Vmn‖p < ∞ for some p ≥ 1.

(i) If p ≤ 2 and α ∧ β > p−1, the condition (3.31) is automatic.

(ii) If p > 1 and α ∧ β > 12 , the condition (3.32) is automatic.

By the same method that is used in the proof of Theorem 3.1, we obtain in Theorem 3.7a Marcinkiewicz–Zygmund type SLLN for double arrays of i.i.d. random elements in arbitraryreal separable Banach spaces. We also omit the details. Theorem 3.7 was originally provedby Giang [4, Theorem 1.1] and by Mikosch and Norvaisa [11, Corollary 4.2] using a differentmethod.

Theorem 3.7 Let 1 ≤ p < 2 and let {Vmn, m ≥ 1, n ≥ 1} be a double array of i.i.d. randomelements in a real separable Banach space with E(‖V11‖p log+ ‖V11‖) < ∞. Then

Smn

(mn)1p

P→ 0 as m ∨ n → ∞ (3.33)

if and only ifSmn

(mn)1p

→ 0 a.s. as m ∨ n → ∞. (3.34)

Remark 3.8 In the one-dimensional case, for i.i.d. random elements {Vn, n ≥ 1}, de Acosta[1, Theorem 3.1] showed that under the condition E‖V1‖p < ∞ where 1 ≤ p < 2, the WLLNimplies the SLLN with norming sequence {n 1

p , n ≥ 1}. This is no longer valid in the multi-dimensional case. To see this, consider a double array of i.i.d. symmetric real-valued randomvariables {Xmn, m ≥ 1, n ≥ 1} with E|X11|p < ∞ and E(|X11|p log+ |X11|) = ∞ for some1 ≤ p < 2. Let Smn =

∑mi=1

∑nj=1 Xij , m ≥ 1, n ≥ 1. Then by Rosalsky and Thanh [16, The-

orem 3.2], we obtain the WLLN (3.33). However, by Gut [5, Theorem 3.2], the correspondingSLLN (3.34) does not hold.

4 Applications

In this section, we will apply the main results to obtain SLLNs for double arrays of independentrandom elements in a real separable Rademacher type p (1 ≤ p ≤ 2) Banach space. Thefollowing theorem, which is a Kolmogorov type SLLN, is a part of Theorem 3.1 of Rosalskyand Thanh [15] (see also Thanh [19, Theorem 2.1] for the real-valued random variable case).However, the proof we present here is entirely different.


Theorem 4.1 Let 1 ≤ p ≤ 2 and let X be a real separable Rademacher type p Banach space.Let {Vmn, m ≥ 1, n ≥ 1} be a double array of independent mean 0 random elements in X . If

∞∑

m=1

∞∑

n=1

E‖Vmn‖p

mαpnβp< ∞, (4.1)

where α > 0, β > 0, then the SLLN

limm∨n→∞

Smn

mαnβ= 0 a.s. (4.2)

holds.

Proof By Theorem 3.1(i), it suffices to show that

Smn

mαnβ

P−→ 0 as m ∨ n → ∞. (4.3)

Since X is of Rademacher type p, it follows from (2.1) that

E

∥∥∥∥

Smn

mαnβ

∥∥∥∥

p

≤ C

mαpnβp

m∑

i=1

n∑

j=1

E‖Vij‖p → 0 as m ∨ n → ∞.

By (4.1) and the Kronecker lemma for double series (see, e.g., Moricz [12, Theorem 1]), andnoting that the summands in (4.1) are nonnegative. Hence, (4.3) follows. The proof is com-plete. �

The following two theorems can be proved by the same method. We omit the details.Theorem 4.2 and Theorem 4.5 are, respectively, a Brunk–Chung type and a Marcinkiewicz–Zygmund type SLLN for double arrays of independent random elements in Rademacher typep Banach spaces. Theorem 4.5 was originally obtained by Giang [4, Theorem 1.2] using adifferent method of proof. Theorem 4.5 will follow immediately from Rosalsky and Thanh [15,Corollary 3.2] if hypothesis that X is of Rademacher type p is strengthened to X being ofRademacher type q for some q ∈ (p, 2].

Theorem 4.2 Let q ≥ 1, 1 ≤ p ≤ 2 and let X be a real separable Rademacher type p Banachspace. Let {Vmn, m ≥ 1, n ≥ 1} be a double array of independent mean 0 random elements inX . If

∞∑

m=1

∞∑

n=1

E‖Vmn‖pq

mαpq−q+1nβpq−q+1< ∞,

where α > 0, β > 0, then the SLLN (4.2) holds.

Corollary 4.3 Let α > 0, β > 0, 1 ≤ p ≤ 2 and let X be a real separable Rademacher typep Banach space. Let {Vmn, m ≥ 1, n ≥ 1} be a double array of independent mean 0 randomelements in X . Assume that for some q ≥ 1, some ε > 0, and all m ≥ 1, n ≥ 1,

E‖Vmn‖pq ≤ C(mαp−1nβp−1)q

((log(m + 1))(log(n + 1))

)1+ε . (4.4)

Then the SLLN (4.2) holds.

Proof The proof is similar to that of Corollary 3.5. �

Remark 4.4 Suppose that 1 ≤ p ≤ 2 and q ≥ 1 are such that supm≥1,n≥1 E‖Vmn‖pq < ∞.If α ∧ β > p−1, then the condition (4.4) is automatic.


Theorem 4.5 Let 1 ≤ p < 2 and let X be a real separable Rademacher type p Banachspace. Let {Vmn, m ≥ 1, n ≥ 1} be a double array of i.i.d. mean 0 random elements in X . IfE(‖V11‖p log+ ‖V11‖) < ∞, then the SLLN

limm∨n→∞

Smn

(mn)1p

= 0 a.s.

holds.

5 An Interesting Example

The following example of Rosalsky and Thanh [15, Example 4.1] demonstrates that Theorem 4.1can fail for 1 < p ≤ 2 if X is not of Rademacher type p or if X is of Rademacher type p but thedouble series of (4.1) diverges. Apropos of Theorem 3.1, the example then also shows that (3.2)and (3.3) can both fail when (3.1) holds. However, it follows from Theorem 4.1 (and also fromRosalsky and Thanh [15, Theorem 3.2]) that if (3.1) holds with p = 1, then (3.2) and (3.3)both hold.

Example 5.1 Let 1 ≤ q < p ≤ 2 and consider the real separable Banach space �q consisting ofabsolute q-th power summable real sequences v = {vk, k ≥ 1} with norm ‖v‖ = (

∑∞k=1 |vk|q)

1q .

Let v(k) denote the element of �q having 1 in its k-th position and 0 elsewhere, k ≥ 1. Letϕ : N × N → N be a one-to-one and onto mapping. Let {Vmn, m ≥ 1, n ≥ 1} be a double arrayof independent random elements in �q by requiring the {Vmn, m ≥ 1, n ≥ 1} to be independentwith

P (Vmn = v(ϕ(m,n))) = P (Vmn = −v(ϕ(m,n))) =12, m ≥ 1, n ≥ 1.

Let α = β = 1q . It is well known that �q is not of Rademacher type p. Note that

∞∑

m=1

∞∑

n=1

E‖Vmn‖p

mαpnβp=

∞∑

m=1

∞∑

n=1

1

mpq n

pq

< ∞

since pq > 1, and so (4.1) holds but (4.2) fails since for all m ≥ 1, n ≥ 1,

‖Smn‖mαnβ

=(mn)

1q

m1q n

1q

= 1 a.s.. (5.1)

Now it is also well known that �q is of Rademacher type q. However,∞∑

m=1

∞∑

n=1

E‖Vmn‖q

mαqnβq=

∞∑

m=1

∞∑

n=1

1mn

= ∞

and we see from (5.1) that (4.2) fails.

References[1] de Acosta, A.: Inequalities for B-valued random vectors with applications to the strong law of large

numbers. Ann. Probab., 9, 157–161 (1981)

[2] Etemadi, N.: Tail probabilities for sums of independent Banach space valued random variables. Sankhya

Ser. A, 47, 209–214 (1985)

[3] Etemadi, N.: Maximal inequalities for partial sums of independent random vectors with multi-dimensional

time parameters. Comm. Statist. Theory Methods, 20, 3909–3923 (1991)

[4] Giang, N. V.: Marcinkiewicz–Zygmund laws for Banach space valued random variables with multidi-

mensional parameters (in Russian). Teor. Veroyatn. Primen., 40, 213–219; English translation in Theory

Probab. Appl., 40, 175–181 (1995)


[5] Gut, A.: Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with

multidimensional indices. Ann. Probab., 6, 469–482 (1978)

[6] Gut, A.: Probability: A Graduate Course, Springer-Verlag, New York, 2005

[7] Hoffmann-Jørgensen, J.: Sums of independent Banach space valued random variables. Studia Math., 52,

159–186 (1974)

[8] Hoffmann-Jørgensen, J., Pisier, G.: The law of large numbers and the central limit theorem in Banach

spaces. Ann. Probab., 4, 587–599 (1976)

[9] Klesov, O. I.: Limit Theorems for Multiple Sums of Independent Random Variables, Springer, Berlin, 2014,

to appear

[10] Loeve, M.: Probability Theory I, 4th ed., Springer-Verlag, New York, 1977

[11] Mikosch, T., Norvaisa, R.: Strong laws of large numbers for fields of Banach space valued random variables.

Probab. Theory Related Fields, 74, 241–253 (1987)

[12] Moricz, F.: The Kronecker lemmas for multiple series and some applications. Acta Math. Acad. Sci. Hun-

gar., 37, 39–50 (1981)

[13] Pyke, R.: Partial sums of matrix arrays, and Brownian sheets. In Stochastic Analysis: A Tribute to the

Memory of Rollo Davidson (Kendall, D. G., Harding, E. F. eds.), John Wiley, London, 1973, 331–348

[14] Rosalsky, A., Thanh, L. V.: Strong and weak laws of large numbers for double sums of independent random

elements in Rademacher type p Banach spaces. Stoch. Anal. Appl., 24, 1097–1117 (2006)

[15] Rosalsky, A., Thanh, L. V.: On almost sure and mean convergence of normed double sums of Banach space

valued random elements. Stoch. Anal. Appl., 25, 895–911 (2007)

[16] Rosalsky, A., Thanh, L. V.: Weak laws of large numbers of double sums of independent random elements

in Rademacher type p and stable type p Banach spaces. Nonlinear Anal., 71, e1065–e1074 (2009)

[17] Stadtmuller, U., Thanh, L. V.: On the strong limit theorems for double arrays of blockwise M -dependent

random variables. Acta Math. Sin., Engl. Series, 27, 1923–1934 (2011)

[18] Taylor, R. L.: Stochastic Convergence of Weighted Sums of Random Elements in Linear Spaces, Lecture

Notes in Mathematics Vol. 672, Springer-Verlag, Berlin, 1978

[19] Thanh, L. V.: Strong law of large numbers and Lp-convergence for double arrays of independent random

variables. Acta Math. Vietnam., 30, 225–232 (2005)

on the laws of large numbers for double arrays of independent random elements in banach spaces

Documents