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

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 Pm i=1 Pn 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, BrunkChung, and MarcinkiewiczZygmund type strong laws of large numbers for double sums in Rademacher type p (1 ≤ p ≤ 2) Banach spaces.

On the Laws of Large Numbers for Double Arrays

of Independent Random Elements in Banach

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 the double sumsPm

i=1

Pn j=1Vij, 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

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 the weak law of large numbers (WLLN) for the double sums Pm

i=1

Pn j=1Vij are equivalent Such double sums differ substantially from the partial sumsPn

i=1Vi, 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.

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

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 EkV k < ∞, then (see, e.g., Taylor [3], p 40) V has an expected value But the expected value can exist when EkV k = ∞ 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

E

n

X

j=1

Vj

p

≤ C

n

X

j=1

E||Vj||p (2.1)

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

S(m, n) = Smn=

m

X

i=1

n

X

j=1

Vij, m ≥ 1, n ≥ 1

For sums of independent random elements, the first lemma provides in (2.2) and (2.3) a Marcinkiewicz-Zygmund 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

E k

n

X

j=1

Vjk − Ek

n

X

j=1

Vjk

p

≤ Cp

n

X

j=1

EkVjkp, for 1 ≤ p ≤ 2, (2.2)

and

E

k

n

X

j=1

Vjk − Ek

n

X

j=1

Vjk

p

≤ Cp

 n

X

j=1

EkVjk2
p/2

+

n

X

j=1

EkVjkp, for p > 2 (2.3)

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

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

P (k

n

X

j=1

Vjk > 2t + s) ≤ 4P2(k

n

X

j=1

Vjk > t) + P ( max

1≤j≤nkVjk > s)

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≤nkSklk > t) ≤ CP (kSmnk > 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

sup

m≥1,n≥1

m+1

X

i=1

n+1

X

j=1

amnij≤ C and 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

lim

m∨n→∞

m+1

X

i=1

n+1

X

j=1

amnijxij = 0

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

∞

X

m=1

∞

X

n=1

EkVmnkp< ∞,

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 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

∞

X

m=1

∞

X

n=1

EkVmnkp

mαpnβp < ∞ for some 1 ≤ p ≤ 2 (3.1) Then

Smn

mαnβ

P

→ 0 as m ∨ n → ∞ (3.2)

if and only if

Smn

mαnβ → 0 a.s as m ∨ n → ∞ (3.3) (ii) Assume that

∞

X

m=1

∞

X

n=1

EkVmnk2p

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 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 V0 = {Vmn0 , m ≥ 1, n ≥ 1}

be two double arrays of independent random elements in a real separable Banach space such that V and V0 are independent copies of each other Then

Smn

mαnβ → 0 a.s as m ∨ n → ∞ (3.5)

if and only if

Pm

i=1

Pn

j=1(Vij− Vij0)

mαnβ → 0 a.s as m ∨ n → ∞ and Smn

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

Smn0 =

m

X

i=1

n

X

j=1

Vij0, m ≥ 1, n ≥ 1

Then for all m ≥ 1, n ≥ 1, kSmnk/(mαnβ) and kSmn0 k/(mαnβ) are i.i.d real valued random variables Let µmn denote a median of kSmnk/(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 [12, p 134]), we have for all ε > 0

P sup

k≤m∨n≤l

kSmn/(mαnβ)k − µmn

> ε

≤ 2P sup

k≤m∨n≤l

kSmn/(mαnβ)k − kSmn0 /(mαnβ)k

> ε

≤ 2P sup

m∨n≥k

Smn− Smn0

mαnβ > ε

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

(3.8)

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

kSmn/(mαnβ)k − µmn

> ε→ 0 as k → ∞ This means 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 kVmnk ≤ mαnβ a.s., m ≥ 1, n ≥ 1 then Smn/(mαnβ) 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 kVmnk ≤ mαnβ a.s for all

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≤nkVijk, m ≥ 1, n ≥ 1 Since

kVmnk ≤ 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 (kSmnk ≥ mαnβε) ≤ 1

8 × 3p (3.13) Now for all A > 0,

Z A

0

tp−1P (kSmnk > mαnβt)dt = 3p

Z A/3 0

tpP (kSmnk > 3mαnβt)dt

≤ 3ph4

Z A/3

0

tp−1P2(kSmnk > mαnβt)dt +

Z A/3 0

tp−1P (Kmn> mαnβt)dti(by Lemma 2.2)

≤4(3ε)

p

p +

1 2

Z A/3 ε

tp−1P (kSmnk > mαnβt)dt + 3p

Z A/3 0

tp−1P (Kmn> mαnβt)dt (by (3.13))

≤4(3ε)

p

p +

1 2

Z A 0

tp−1P (kSmnk > mαnβt)dt + 3p

Z 1 0

tp−1P (Kmn> mαnβt)dt (by (3.12))

(3.14)

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

Z A

0

tp−1P (kSmnk > mαnβt)dt ≤ 8(3ε)

p

p + 2 × 3

pZ 1

0

tp−1P (Kmn> mαnβt)dt and hence

E Smn

mαnβ

p

= p

Z ∞ 0

tp−1P (kSmnk > mαnβt)dt

= p lim

A→∞

Z A 0

tp−1P (kSmnk > mαnβt)dt

≤ 8(3ε)p+ (2p)3p

Z 1 0

tp−1P (Kmn> mαnβt)dt

(3.15)

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

P (Kmn> mαnβt) ≤ P ( max

k≤m,l≤nkSklk > mαnβt/4)

≤ CP (kSmnk > mαnβt/(4C)) → 0 as m ∨ n → ∞

(3.16)

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

Z 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 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

mαnβ → 0 a.s as m ∨ n → ∞ (3.18)

if and only if

P2 m −1 i=2 m−1

P2 n −1 j=2 n−1Vij

2mα2nβ → 0 a.s as m ∨ n → ∞ (3.19) Proof Let

Tmn=

2m−1

X

i=2 m−1

2n−1

X

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

∞

X

m=1

∞

X

n=1

P (kTmnk > 2mα2nβε) < ∞ for all ε > 0 (3.20)

Mrs= max

2 r−1 ≤u≤2 r −1,2 s−1 ≤v≤2 s −1

u

X

i=2 r−1

v

X

j=2 s−1

Vij , r ≥ 1, s ≥ 1

By Lemma 2.3 and (3.20), we have

∞

X

r=1

∞

X

s=1

P (Mrs> 2rα2sβε) ≤ C

∞

X

r=1

∞

X

s=1

P (kTrsk > 2rα2sβε/C) < ∞ for all ε > 0

This ensures that

Mrs

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,

kSmnk ≤

k+1

X

r=1

l+1

X

s=1

Mrs

and so

Smn

mαnβ ≤

k+1

X

r=1

l+1

X

s=1

2rα2sβ

2kα2lβ Mrs

2rα2sβ (3.22) Note that

sup

k≥1,l≥1

k+1

X

r=1

l+1

X

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

X

r=1

l+1

X

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 the theorem assuming the {Vmn, m ≥ 1, n ≥ 1} are symmetric Set

Wmn= VmnI(kVmnk ≤ mαnβ), m ≥ 1, n ≥ 1

By Markov’s inequality and (3.1)

∞

X

n=1

∞

X

m=1

P kVmnk > mαnβ
≤

∞

X

n=1

∞

X

m=1

EkVmnkp

mαpnβp < ∞ (3.25)

Also by (3.1),

∞

X

n=1

∞

X

m=1

EkWmnkp

mαpnβp < ∞ (3.26)

By (3.25) and the Borel-Cantelli lemma, it suffices to prove

Pm i=1

Pn j=1Wij

mαnβ → 0 a.s as m ∨ n → ∞ (3.27) Using (3.25) and the Borel-Cantelli lemma again, it follows from (3.2) that

Pm i=1

Pn j=1Wij

mαnβ

P

→ 0 as m ∨ n → ∞

Thus, Lemma 3.3 ensures that

Pm i=1

Pn j=1Wij

mαnβ → 0 in L1 as m ∨ n → ∞ and so

P2 m −1

i=2 m−1

P2 n −1 j=2 n−1Wij

2mα2nβ =

P2 m −1 i=1

P2 n −1 j=1 Wij

2mα2nβ −

P2 m −1 i=1

P2 n−1 −1 j=1 Wij

2β2mα2(n−1)β

−

P2m−1−1 i=1

P2n−1 j=1 Wij

2α2(m−1)α2nβ +

P2m−1−1 i=1

P2n−1−1 j=1 Wij

2α2β2(m−1)α2(n−1)β

→ 0 in L1 as m ∨ n → ∞

(3.28)

Now if we can show that

kP2 m −1

i=2 m−1

P2 n −1 j=2 n−1Wijk − EkP2 m −1

i=2 m−1

P2 n −1 j=2 n−1Wijk

2mα2nβ → 0 a.s as m ∨ n → ∞, (3.29) then it follows from (3.28) that

P2m−1 i=2 m−1

P2n−1 j=2 n−1Wij

2mα2nβ → 0 a.s as m ∨ n → ∞ which yields (3.27) via Lemma 3.4 To prove (3.29), note that

∞

X

m=1

∞

X

n=1

E

kP2 m −1 i=2 m−1

P2 n −1 j=2 n−1Wijk − EkP2 m −1

i=2 m−1

P2 n −1 j=2 n−1Wijk

p

2mαp2nβp

≤ C

∞

X

m=1

∞

X

n=1

P2m−1 i=2 m−1

P2n−1 j=2 n−1EkWijkp

2mαp2nβp (by (2.2) of Lemma 2.1)

≤ C

∞

X

k=1

∞

X

l=1

EkWklkp

kαplβp < ∞ (by (3.26))

(3.30)

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)

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

EkVmnkp≤ C m

αp−1nβp−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

EkVmnk2p≤ C m

p(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)

∞

X

m=1

∞

X

n=1

EkVmnkp

mαpnβp ≤ C

∞

X

m=1

∞

X

n=1

1 m(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≥1EkVmnkp< ∞ 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(kV11kplog+kV11k) < ∞ Then

Smn

(mn)1/p

P

→ 0 as m ∨ n → ∞ (3.33)

if and only if

Smn (mn)1/p → 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 [5, Theorem 3.1] showed that under the condition EkV1kp < ∞ where 1 ≤ p < 2, the WLLN implies the SLLN with norming sequence {n1/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 random variables {Xmn, m ≥ 1, n ≥ 1} with E|X11|p< ∞ and E(|X11|plog+|X11|) = ∞ for some 1 ≤ p < 2 Let Smn = Pm

i=1

Pn j=1Xij, 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

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

∞

X

m=1

∞

X

n=1

E||Vmn||p

mαpnβp < ∞, (4.1) where α > 0, β > 0, then the SLLN

lim

m∨n→∞

Smn

mαnβ = 0 a.s (4.2) obtains

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

X

i=1

n

X

j=1

EkVijkp → 0 as m ∨ n → ∞

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

∞

X

m=1

∞

X

n=1

E||Vmn||pq

mαpq−q+1nβpq−q+1 < ∞, where α > 0, β > 0, then the SLLN (4.2) obtains

In Theorem 3.1, condition (3.1) is a Kolmogorov type condition for the SLLN for double. .. class="page_container" data-page="10">

4 Applications

In this section, we will apply the main results to obtain SLLNs for double arrays of independent random elements in a real... ∞ (3.17) The conclusion (3.11) follows from (3.15), (3.17), and the arbitrariness of ε >

We use the L´evy type inequality for double arrays of independent symmetric random elements