On the Laws of Large Numbers for Double Arrays of Independent Random Elements in Banach Spaces

On the Laws of Large Numbers for Double Arrays
of Independent Random Elements in Banach
Spaces ∗
Andrew ROSALSKY, Le Van THANH, Nguyen Thi THUY

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
n
the double sums m
i=1
j=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.
Key Words and Phrases: 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.
2010 Mathematics Subject Classifications: 60F05, 60F15, 60B11, 60B12

1

Introduction

Throughout this paper, we consider a double array {Vmn , m ≥ 1, n ≥ 1} of independent random
elements defined on a probability space (Ω, F, P ) and taking values in a real separable Banach space
X with norm || · ||. We provide conditions under which the strong law of large numbers (SLLN) and
m
n
the weak law of large numbers (WLLN) for the double sums i=1 j=1 Vij are equivalent. Such

n
double sums differ substantially from the partial sums i=1 Vi , n ≥ 1 of a sequence 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 identically distributed (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 underlying Banach space. We then apply
the main results to obtain Kolmogorov, Brunk-Chung, and Marcinkiewicz-Zygmund type SLLNs for
double sums in Rademacher type p (1 ≤ p ≤ 2) Banach spaces.
While in the current work attention is restricted to considering double sums, the results can of
course be extended by the same method to multiple sums over lattice points of any dimension.
∗ The research of the second author was supported by the Vietnam Institute for Advanced Study in Mathematics
(VIASM) and the Vietnam National Foundation for Sciences and Technology Development (NAFOSTED) under
grant number 101.01.2012.13. The research of the third author was supported by the Vietnam National Foundation
for Sciences and Technology Development (NAFOSTED) under grant number 101.03.2012.17.

1


The reader may refer to Rosalsky and Thanh [1] for a brief discussion of a historical nature
concerning double sums and on their importance in the field of statistical physics. For the case of
i.i.d. real valued random variables, a major surrey article concerning double sums was prepared by
Pyke [2]. In Pyke [2], he discussed fluctuation theory, the limiting Brownian sheet, the SLLN, and
various other limit theorems. Currently, Professor Oleg I. Klesov (National Technical University of
Ukraine) is preparing a comprehensive book on multiple sums of independent random variables.
The plan of the paper is as follows. Notation, technical definitions, and five known lemmas which
are used in proving the main results are consolidated into Section 2. In Section 3, we establish the
main results after first proving three new lemmas. The applications of the main results are presented
in Section 4. Section 5 contains an example pertaining to Theorems 3.1 and 4.1.

2

Preliminaries

In this section, notation, technical definitions, and lemmas which are needed in connection with the
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 not necessarily the
same one in each appearance.
The expected value or mean of an X -valued random element V , denoted EV , is defined to be the
Pettis integral provided it exists. If E V < ∞, then (see, e.g., Taylor [3], p. 40) V has an expected
value. But the expected value can exist when E V = ∞. For an example, see Taylor [3], p. 41.
Hoffmann-Jørgensen and Pisier [4] proved for 1 ≤ p ≤ 2 that a real separable Banach space is of
Rademacher type p if and only if there exists a constant C such that
p

n

n

E||Vj ||p

≤C

Vj

E
j=1

(2.1)

j=1

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

n

Vij , m ≥ 1, n ≥ 1.

S(m, n) = Smn =
i=1 j=1

For sums of independent random elements, the first lemma provides in (2.2) and (2.3) a MarcinkiewiczZygmund type inequality and a Rosenthal type inequality, respectively. Lemma 2.1 is due to de
Acosta [5, Theorem 2.1].
Lemma 2.1. Let {Vj , 1 ≤ j ≤ n} be a collection of n independent random elements. Then for every
p ≥ 1, there is a positive constant Cp < ∞ depending only on p such that
n

n

Vj − E

E
j=1

and

n

j=1

≤ Cp

j=1

n

Vj − E

E

n

p

Vj

j=1

p

, for 1 ≤ p ≤ 2,

(2.2)

j=1

n

p

Vj

E Vj

n

≤ Cp

E Vj
j=1

2 p/2

+

E Vj

p

, for p > 2.

j=1

The following lemma is due to Hoffmann-Jørgensen [6]; see the proof of Theorem 3.1 of [6].
2

(2.3)

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

n
2

Vj > 2t + s) ≤ 4P (

P(
j=1

Vj > t) + P ( max Vj > s).
1≤j≤n

j=1

The next lemma is L´evy’s inequality for double arrays of independent symmetric random elements
in Banach spaces. It is due to Etemadi [7, Corollary 1.2]. We note that Etemadi [7] 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 symmetric random
elements in a real separable Banach space. Then there exist a constant C such that for all t > 0,
P ( max

k≤m,l≤n

Skl > t) ≤ CP ( Smn > t/C).

The following result is a double sum analogue of the Toeplitz lemma (see, e.g., Lo`eve [8], p. 250)

and is due to Stadtm¨
uller and Thanh [9, Lemma 2.2].
Lemma 2.4. Let {amnij , 1 ≤ i ≤ m + 1, 1 ≤ j ≤ n + 1, m ≥ 1, n ≥ 1} be an array of positive
constants such that
m+1 n+1

amnij ≤ C and

sup
m≥1,n≥1 i=1 j=1

lim

m∨n→∞

amnij = 0 for every fixed i, j.

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

m∨n→∞

xmn = 0,

then
m+1 n+1

amnij xij = 0.

lim

m∨n→∞

i=1 j=1

The last lemma in this section has an easy proof; see Rosalsky and Thanh [10, Lemma 2.1].
Lemma 2.5. Let {Vmn , m ≥ 1, n ≥ 1} be a double array of random elements in a real separable
Banach space and let p > 0. If
∞

∞

E Vmn

p

< ∞,

m=1 n=1

then
Vmn → 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 divergence half, it
is assumed that the array {Amn , m ≥ 1, n ≥ 1} is comprised of independent events.

3

3

Main Results

With the preliminaries accounted for, the first main result may be established. Theorem 3.1 considers
the independent but nonidentically distributed case while Theorem 3.7 considers the i.i.d. case. In
these theorems, no assumptions are made concerning the geometry of the underlying Banach space.
In Theorem 3.1, condition (3.1) is a Kolmogorov type condition for the SLLN for double arrays
whereas condition (3.4) is a Brunk-Chung type condition for 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 independent
random elements in a real separable Banach space.
(i) Assume that
∞ ∞
E Vmn p
< ∞ for some 1 ≤ p ≤ 2.
(3.1)
mαp nβp
m=1 n=1
Then

Smn P
→ 0 as m ∨ n → ∞
m α nβ

(3.2)

Smn
→ 0 a.s. as m ∨ n → ∞.
mα nβ

(3.3)

E Vmn 2p
2αp+1−p
m
n2βp+1−p

(3.4)

if and only if

(ii) Assume that
∞

∞

m=1 n=1

< ∞ for some p > 1.

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. Some of
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 random elements. Lemma
3.2 is a double sum analogue (with more general norming constants) of Lemma 1 of Etemadi [11].
Lemma 3.2. Let α > 0, β > 0 and let V = {Vmn , m ≥ 1, n ≥ 1} and V = {Vmn , m ≥ 1, n ≥ 1}
be two double arrays of independent random elements in a real separable Banach space such that V
and V are independent copies of each other. Then
Smn
→ 0 a.s. as m ∨ n → ∞

m α nβ

(3.5)

if and only if
m
i=1

n
j=1 (Vij
mα nβ

− Vij )

→ 0 a.s. as m ∨ n → ∞ and

Smn P
→ 0 as m ∨ n → ∞.
mα nβ

(3.6)

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

n

Vij , m ≥ 1, n ≥ 1.

Smn =

i=1 j=1

Then for all m ≥ 1, n ≥ 1, Smn /(mα nβ ) and Smn /(mα nβ ) are i.i.d. real valued random
variables. Let µmn denote a median of Smn /(mα nβ ), m ≥ 1, n ≥ 1. By the second half of (3.6),
µmn → 0 as m ∨ n → ∞.
4

(3.7)


By the strong symmetrization inequality (see, e.g., Gut [12, p. 134]), we have for all ε > 0
P

Smn /(mα nβ ) − µmn > ε

sup
k≤m∨n≤l

≤ 2P

sup
k≤m∨n≤l

≤ 2P

sup
m∨n≥k

Smn /(mα nβ ) − Smn /(mα nβ )

>ε
(3.8)

Smn − Smn
mα nβ

>ε

→ 0 as k → ∞ (by the first half of (3.6)).
Letting l → ∞ in (3.8), we have P supm∨n≥k Smn /(mα nβ ) − µmn > ε → 0 as k → ∞. This
means that
Smn
− µmn → 0 a.s. as m ∨ n → ∞.
(3.9)
m α nβ
By combining (3.7) and (3.9), we obtain (3.5).
The second lemma shows that if Vmn ≤ mα nβ a.s., m ≥ 1, n ≥ 1 then Smn /(mα nβ ) obeying
the WLLN as m ∨ n → ∞ is indeed equivalent to its convergence in Lp to 0 as m ∨ n → ∞ for any
p > 0.
Lemma 3.3. Let α > 0, β > 0 and let {Vmn , m ≥ 1, n ≥ 1} be a double array of independent
symmetric random elements in a real separable Banach space such that Vmn ≤ mα nβ a.s. for all
m ≥ 1, n ≥ 1. If
Smn P
→ 0 as m ∨ n → ∞,
(3.10)
m α nβ
then for all p > 0,
Smn
→ 0 in Lp as m ∨ n → ∞.
(3.11)

m α nβ
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 ≤ mα nβ a.s. for all m ≥ 1, n ≥ 1,
Kmn ≤ mα nβ 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 ≥ mα nβ ε) ≤

1
.
8 × 3p

(3.13)

Now for all A > 0,
A

A/3

tp−1 P ( Smn > mα nβ t)dt = 3p
0

tp P ( Smn > 3mα nβ t)dt
0

A/3

≤ 3p 4

A/3

tp−1 P 2 ( Smn > mα nβ t)dt +
0
p

≤
≤

4(3ε)
1
+
p
2
4(3ε)p
1
+
p
2

tp−1 P (Kmn > mα nβ t)dt (by Lemma 2.2)
0

A/3

A/3

tp−1 P ( Smn > mα nβ t)dt + 3p
ε

tp−1 P (Kmn > mα nβ t)dt (by (3.13))
0
1

A

tp−1 P ( Smn > mα nβ t)dt + 3p
0

tp−1 P (Kmn > mα nβ t)dt (by (3.12)).
0

(3.14)
5


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

tp−1 P ( Smn > mα nβ t)dt ≤
0

and hence
E

Smn
m α nβ

8(3ε)p

+ 2 × 3p
p

1

tp−1 P (Kmn > mα nβ t)dt
0

∞

p

tp−1 P ( Smn > mα nβ t)dt

=p
0

A

tp−1 P ( Smn > mα nβ t)dt

= p lim

A→∞

(3.15)

0
1

tp−1 P (Kmn > mα nβ t)dt.

≤ 8(3ε)p + (2p)3p
0

Note that for all m ≥ 1, n ≥ 1, Kmn ≤ 4 maxk≤m,l≤n Skl and so by Lemma 2.3 and (3.10) we have
for t > 0
P (Kmn > mα nβ t) ≤ P ( max Skl > mα nβ t/4)
k≤m,l≤n
(3.16)
≤ CP ( Smn > mα nβ t/(4C)) → 0 as m ∨ n → ∞.
Hence by the Lebesgue dominated convergence theorem, (3.16) implies that
1

tp−1 P (Kmn > mα nβ t)dt → 0 as m ∨ n → ∞.

(3.17)

0

The conclusion (3.11) follows from (3.15), (3.17), and the arbitrariness of ε > 0.
We use the L´evy type inequality for double arrays of independent symmetric random elements
(Lemma 2.3) as a key tool to prove the following lemma. This lemma is a double sum version of
Lemma 3.2 of de Ascota [5].
Lemma 3.4. Let α > 0, β > 0 and let {Vmn , m ≥ 1, n ≥ 1} be a double array of independent
symmetric random elements in a real separable Banach space. Then
Smn
→ 0 a.s. as m ∨ n → ∞
mα nβ
if and only if

2m −1
2n −1
i=2m−1
j=2n−1
mα
2 2nβ

Vij

→ 0 a.s. as m ∨ n → ∞.

(3.18)

(3.19)

Proof. Let
2m −1

2n −1

Vij , m ≥ 1, n ≥ 1.

Tmn =
i=2m−1 j=2n−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 of independent
random elements, by the Borel-Cantelli lemma

∞

∞

P ( Tmn > 2mα 2nβ ε) < ∞ for all ε > 0.
m=1 n=1

6

(3.20)


Let

u

Mrs =

v

Vij , r ≥ 1, s ≥ 1.

max

2r−1 ≤u≤2r −1,2s−1 ≤v≤2s −1

i=2r−1 j=2s−1

By Lemma 2.3 and (3.20), we have
∞

∞

∞

∞

P (Mrs > 2rα 2sβ ε) ≤ C
r=1 s=1

P ( Trs > 2rα 2sβ ε/C) < ∞ for all ε > 0.
r=1 s=1

This ensures that

Mrs
→ 0 a.s. as r ∨ s → ∞.
2rα 2sβ
For m ≥ 1, n ≥ 1, let k ≥ 0, l ≥ 0 be such that

(3.21)

2k ≤ m ≤ 2k+1 − 1 and 2l ≤ n ≤ 2l+1 − 1.
Then for m ≥ 1, n ≥ 1,
k+1 l+1

Smn ≤

Mrs
r=1 s=1

and so
Smn
mα nβ

k+1 l+1

≤
r=1 s=1

2rα 2sβ Mrs
.
.
2kα 2lβ 2rα 2sβ

(3.22)

Note that
k+1 l+1

sup
k≥1,l≥1 r=1 s=1

2rα 2sβ
< ∞, and
2kα 2lβ

2rα 2sβ
= 0 for every fixed r, s.
k∨l→∞ 2kα 2lβ

lim

(3.23)

Hence from (3.21) and (3.23), we get by applying Lemma 2.4 that
k+1 l+1

r=1 s=1

2rα 2sβ Mrs
.
→ 0 a.s. as k ∨ l → ∞.
2kα 2lβ 2rα 2sβ

(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 the
theorem assuming the {Vmn , m ≥ 1, n ≥ 1} are symmetric. Set
Wmn = Vmn I( Vmn ≤ mα nβ ), m ≥ 1, n ≥ 1.
By Markov’s inequality and (3.1)
∞

∞

∞

P

Vmn > mα nβ ≤

n=1 m=1

∞

E Vmn p
< ∞.
mαp nβp
n=1 m=1

(3.25)

Also by (3.1),
∞

∞

E Wmn p
< ∞.
mαp nβp
n=1 m=1
7

(3.26)


By (3.25) and the Borel-Cantelli lemma, it suffices to prove
m
i=1

n
j=1
m α nβ

Wij

→ 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

n
j=1
m α nβ

Wij

P

→ 0 as m ∨ n → ∞.

Thus, Lemma 3.3 ensures that
m
i=1

n
j=1

m α nβ

Wij

=

2m −1
2n −1
i=1
j=1
2mα 2nβ

→ 0 in L1 as m ∨ n → ∞

and so
2m −1
2n −1
i=2m−1
j=2n−1
2mα 2nβ

Wij

−

Wij

2m −1
2n−1 −1
Wij

i=1
j=1
2β 2mα 2(n−1)β

−

2m−1 −1
2n −1
i=1
j=1 Wij
α
(m−1)α
2 2
2nβ

+

2m−1 −1
2n−1 −1
Wij
i=1
j=1
α
β
(m−1)α
(n−1)β
2 2 2
2

(3.28)

→ 0 a.s. as m ∨ n → ∞,

(3.29)

→ 0 in L1 as m ∨ n → ∞.
Now if we can show that
2m −1
i=2m−1

2n −1
j=2n−1

Wij − E

2m −1
i=2m−1

2n −1
j=2n−1

Wij

2mα 2nβ
then it follows from (3.28) that
2m −1
2n −1
i=2m−1
j=2n−1
2mα 2nβ

Wij

→ 0 a.s. as m ∨ n → ∞

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

∞

2m −1
i=2m−1

E

Wij − E

2m −1
i=2m−1

2n −1
j=2n−1

p

Wij

2mαp 2nβp

m=1 n=1

∞

2n −1
j=2n−1

2m −1
i=2m−1

∞

≤C
m=1 n=1
∞ ∞

≤C
k=1 l=1

2n −1
j=2n−1 E
2mαp 2nβp

Wij

p

(by (2.2) of Lemma 2.1)

(3.30)

E Wkl p

< ∞ (by (3.26)).
k αp lβp

By Lemma 2.5, (3.29) follows from (3.30). The proof of Theorem 3.1 (i) is completed. The proof of
Theorem 3.1 (ii) is similar and we omit the details but we point out that (2.3) of Lemma 2.1 is used
instead of (2.2).
8


Corollary 3.5. Let α > 0, β > 0 and let {Vmn , m ≥ 1, n ≥ 1} be a double array of independent
random elements in a real separable Banach space.
(i) Assume that for some 1 ≤ p ≤ 2, some ε > 0, and all m ≥ 1, n ≥ 1 that
E Vmn

p

≤C

mαp−1 nβp−1
.
((log(m + 1))(log(n + 1)))1+ε

(3.31)

Then (3.2) and (3.3) are equivalent.
(ii) Assume that for some p > 1, some ε > 0 and all m ≥ 1, n ≥ 1 that
E Vmn

2p

≤C

mp(2α−1) np(2β−1)
.
((log(m + 1))(log(n + 1)))1+ε

(3.32)

Then (3.2) and (3.3) are equivalent.
Proof. (i) Note that by (3.31)
∞

∞

∞

∞

E Vmn p
1
≤C
<∞
αp
βp
1+ε
m n
m(log(m + 1)) n(log(n + 1))1+ε
m=1 n=1
m=1 n=1
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 > 1/2, the condition (3.32) is automatic.
By the same method that is used in the proof of Theorem 3.1, we obtain in Theorem 3.7 a
Marcinkiewicz-Zygmund type SLLN for double arrays of i.i.d. random elements in arbitrary real
separable Banach spaces. We also omit the details. Theorem 3.7 was originally proved by Mikosch
and Norvaiˇsa [13, Corollary 4.2] and by Giang [14, Theorem 1.1] using a different method.
Theorem 3.7. Let 1 ≤ p < 2 and let {Vmn , m ≥ 1, n ≥ 1} be a double array of i.i.d. random
elements in a real separable Banach space with E( V11 p log+ V11 ) < ∞. Then
Smn P
→ 0 as m ∨ n → ∞
(mn)1/p

(3.33)

Smn
→ 0 a.s. as m ∨ n → ∞.
(mn)1/p

(3.34)

if and only if

Remark 3.8. In the one dimensional case, for i.i.d. random elements {Vn , n ≥ 1}, de Acosta
[5, Theorem 3.1] showed that under the condition E V1 p < ∞ where 1 ≤ p < 2, the WLLN
implies the SLLN with norming sequence {n1/p , n ≥ 1}. This is no longer valid in the multidimensional case. To see this, consider a double array of i.i.d. symmetric real valued random
variables {Xmn , m ≥ 1, n ≥ 1} with E|X11 |p < ∞ and E(|X11 |p log+ |X11 |) = ∞ for some 1 ≤ p < 2.
m
n
Let Smn = i=1 j=1 Xij , m ≥ 1, n ≥ 1. Then by Theorem 3.2 of Rosalsky and Thanh [15], we

obtain the WLLN (3.33). However, by Theorem 3.2 of Gut [16], the corresponding SLLN (3.34)
does not hold.

9


4

Applications

In this section, we will apply the main results to obtain SLLNs for double arrays of independent
random elements in a real separable Rademacher type p (1 ≤ p ≤ 2) Banach space. The following
theorem, which is a Kolmogorov type SLLN, is a part of Theorem 3.1 of Rosalsky and Thanh [10]
(see also Thanh [17, Theorem 2.1] for the real valued random variables 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
∞

∞

E||Vmn ||p
< ∞,
mαp nβp
m=1 n=1

(4.1)

Smn
= 0 a.s.

mα nβ

(4.2)

where α > 0, β > 0, then the SLLN
lim

m∨n→∞

obtains.
Proof. By Theorem 3.1 (i), it suffices to show that
Smn P
−
→ 0 as m ∨ n → ∞.
mα nβ

(4.3)

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

Smn
mα nβ

p

≤

C
αp

m nβp

m

n

E Vij

p

→ 0 as m ∨ n → ∞

i=1 j=1

by (4.1) and the Kronecker lemma for double series (see, e.g., M´oricz [18, Theorem 1]) noting that
the summands in (4.1) are nonnegative. Hence (4.3) follows. The proof is completed.
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 type p Banach spaces.
Theorem 4.5 was originally obtained by Giang [14, Theorem 1.2] using a different method of proof.
Theorem 4.5 will follow immediately from Corollary 3.2 of Rosalsky and Thanh [10] if hypothesis
that X is of Rademacher type p is strengthened to X being of Rademacher 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 Banach space.
Let {Vmn , m ≥ 1, n ≥ 1} be a double array of independent mean 0 random elements in X . If
∞

∞

E||Vmn ||pq
< ∞,

mαpq−q+1 nβpq−q+1
m=1 n=1
where α > 0, β > 0, then the SLLN (4.2) obtains.

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Corollary 4.3. Let α > 0, β > 0, 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 . Assume that for some q ≥ 1, some ε > 0, and all m ≥ 1, n ≥ 1 that
pq

E Vmn

(mαp−1 nβp−1 )q

≤C

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

1+ε .

(4.4)

Then the SLLN (4.2) obtains.
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
α ∧ β > p−1 , then the condition (4.4) is automatic.

pq

< ∞. If

Theorem 4.5. 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 i.i.d. mean 0 random elements in X . If E( V11 p log+ V11 ) <
∞, then the SLLN
Smn
= 0 a.s.
lim
m∨n→∞ (mn)1/p
obtains.

5

An Interesting Example

The following example of Rosalsky and Thanh [10, Example 4.1] demonstrates that Theorem 4.1
can fail for 1 < p ≤ 2 if X is not of Rademacher type p or if X is of Rademacher type p but the
double 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 from Theorem
3.2 of Rosalsky and Thanh [10]) 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 of
1/q
∞
absolute q th power summable real sequences v = {vk , k ≥ 1} with norm v = ( k=1 |vk |q ) . Let
v (k) denote the element of q having 1 in its k th position and 0 elsewhere, k ≥ 1. Let ϕ : N × N → N
be a one-to-one and onto mapping. Let {Vmn , m ≥ 1, n ≥ 1} be a double array of independent
random elements in q by requiring the {Vmn , m ≥ 1, n ≥ 1} to be independent with
1
, m ≥ 1, n ≥ 1.

2
is not of Rademacher type p. Note that

P Vmn = v (ϕ(m,n)) = P Vmn = −v (ϕ(m,n)) =
Let α = β = 1/q. It is well known that
∞

∞

q

∞

∞

E Vmn p
1
=
<∞
αp nβp
p/q np/q
m
m
m=1 n=1
m=1 n=1
since p/q > 1 and so (4.1) holds but (4.2) fails since for all m ≥ 1, n ≥ 1,

Now it is also well known that

q

∞

Smn
(mn)1/q
= 1/q 1/q = 1 a.s.
α
β
m n
m n
is of Rademacher type q. However,
∞

∞

∞

1
E Vmn q
=
=∞
αq nβq
m
mn
m=1 n=1
m=1 n=1
and we see from (5.1) that (4.2) fails.
11

(5.1)

References
[1] Rosalsky, A., and Thanh, L.V. 2006. Strong and weak laws of large numbers for double sums
of independent random elements in Rademacher type p Banach spaces. Stochastic Anal. Appl.
24:1097–1117.
[2] Pyke, R. 1973. 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, 331–348.
[3] Taylor, R.L. 1978. Stochastic Convergence of Weighted Sums of Random Elements in Linear
Spaces, Lecture Notes in Mathematics Vol. 672. Springer-Verlag, Berlin.
[4] Hoffmann-Jørgensen, J., and Pisier, G. 1976. The law of large numbers and the central limit
theorem in Banach spaces. Ann. Probab. 4:587–599.
[5] de Acosta, A. 1981. Inequalities for B-valued random vectors with applications to the strong
law of large numbers. Ann. Probab. 9:157–161.
[6] Hoffmann-Jørgensen, J. 1974. Sums of independent Banach space valued random variables.
Studia Math. 52:159–186.
[7] Etemadi, N. 1991. Maximal inequalities for partial sums of independent random vectors with
multi-dimensional time parameters. Comm. Statist. Theory Methods 20:3909–3923.
[8] Lo`eve, M. 1977. Probability Theory I, 4th Ed.; Springer-Verlag, New York.
[9] Stadtm¨
uller, U., and Thanh, L. V. 2011. On the strong limit theorems for double arrays of
blockwise M -dependent random variables. Acta Math. Sin. (Engl. Ser.) 27:1923–1934.
[10] Rosalsky, A., and Thanh, L.V. 2007. On almost sure and mean convergence of normed double
sums of Banach space valued random elements. Stochastic Anal. Appl. 25:895–911.
[11] Etemadi, N. 1985. Tail probabilities for sums of independent Banach space valued random
variables. Sankhy¯
a Ser. A 47:209–214.
[12] Gut, A. 2005. Probability: A graduate course. Springer-Verlag, New York.
[13] Mikosch, T., and Norvaiˇsa, R. 1987. Strong laws of large numbers for fields of Banach space
valued random variables. Probab. Theory Related Fields 74:241–253.

[14] Giang, N.V. 1995. Marcinkiewicz-Zygmund laws for Banach space valued random variables with
multidimensional parameters. Teor. Veroyatnost. i Primenen. 40:213–219 (in Russian); English
translation in Theory Probab. Appl. 40:175–181.
[15] Rosalsky, A., and Thanh, L.V. 2009. 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.
[16] Gut, A. 1978. Marcinkiewicz laws and convergence rates in the law of large numbers for random
variables with multidimensional indices. Ann. Probab. 6:469–482.
[17] Thanh, L. V. 2005. Strong law of large numbers and Lp -convergence for double arrays of independent random variables. Acta Math. Vietnam. 30:225–232.
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[18] M´
oricz, F. 1981. The Kronecker lemmas for multiple series and some applications. Acta Math.
Acad. Sci. Hungar. 37:39–50.
Andrew Rosalsky
Department of Statistics
University of Florida
Gainesville
Florida 32611 − 8545, USA
E-mail:

Le Van Thanh and Nguyen Thi Thuy
Department of Mathematics
Vinh University
182 Le Duan, Vinh, Nghe An
Vietnam
Email: ;

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