ON CHUNG-TEICHER TYPE STRONG LAW FOR ARRAYS OF VECTOR-VALUED RANDOM VARIABLES

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PII. S0161171204301031 http://ijmms.hindawi.com

ON CHUNG-TEICHER TYPE STRONG LAW FOR ARRAYS OF VECTOR-VALUED RANDOM VARIABLES

ANNA KUCZMASZEWSKA Received 2 January 2003

We study the equivalence between the weak and strong laws of large numbers for arrays of row-wise independent random elements with values in a Banach spaceᏮ. The conditions under which this equivalence holds are of the Chung or Chung-Teicher types. These conditions are expressed in terms of convergence of speciﬁc series ando(1)requirements on speciﬁc weighted row-wise sums. Moreover, there are not any conditions assumed on the geometry of the underlying Banach space.

2000 Mathematics Subject Classiﬁcation: 60F15, 60B12.

Let(Ω,Ᏺ,P)be a probability space and letᏮbe a real separable Banach space with norm·. A strongly measurable transformation fromΩtoᏮis said to be aᏮ^-valued random variable or a random element. IfEX<∞, then the expected value is deﬁned by the Bochner integral.

Let{Xn, n≥1}be a sequence ofᏮ-valued random variables. Then{Xn, n≥1}is said to obey the strong law of large numbers (SLLN) if there exist sequences of real numbers{an, n≥1}and{bn, n≥1}such that

n j=1

aj Xj−bj

→0 a.s.,n → ∞. (1)

Suﬃcient conditions for SLLN use very often the geometry of a Banach space, that is, they assume thatᏮis a special-type space, for instanceᏮis of Rademacher typep, 1< p≤2.

The spaceᏮis of Rademacher typepif there exists a positive constantCsuch that

E

∞ n=1

εnxn

p

≤C ∞ n=1

xn^p (2)

for each(x1,x2,...)∈C(B), where{εn, n≥1}is a Bernoulli sequence, that is,εn,n≥1, are i.i.d. random variables andP[εn=1]=P[εn= −1]=1/2,C(B)= {(x1,x2,...)∈B^∞: _∞

n=1εnxnconverges in probability},B^∞=B×B×B×···.

The sufficient conditions for SLLN for random elements taking value in a space of Rademacher typepwere presented by Woyczyński [15], Hoffmann-Jørgensen and Pisier [6], Kuczmaszewska and Szynal [8], and Adler et al. [1].

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444 ANNA KUCZMASZEWSKA

The type of Marcinkiewicz-Zygmunt SLLN provides that for 1≤α <2 and a sequence {Xn, n≥1}of i.i.d.Ꮾ-valued random variables,

1 n¹^/α

n i=1

Xi−EXi

→0 a.s.,n→ ∞, (3)

if and only ifEX1<∞and the Banach spaceᏮis of a Rademacher typepforα < p≤2 (cf. [15]).

The classical result of Hoﬀmann-Jørgensen and Pisier [6] proved that the assumption that a Banach spaceᏮis the space of Rademacher typep, 1≤p≤2, is equivalent to the fact that the condition

∞ n=1

EXn^p

n^p <∞ (4)

implies SLLN for a sequence ofᏮ-valued independent random variables{Xn, n≥1} withEXn=0,n≥1.

In view of many statistical applications, it is important to consider the array-type SLLN.

Let{kn, n≥1}be a strictly increasing sequence of positive integers. An array of Ꮾ-valued random variables{Xni, 1≤i≤kn, n≥1}obeys the general array type of SLLN if

kn

i=1

ani

Xni−cni

→0 a.s.,n→ ∞, (5)

where{ani,1≤i≤kn, n≥1}and{cni,1≤i≤kn, n≥1}are suitable arrays of constants (weights) andᏮ-valued elements, respectively, and 0 denotes the zero-element inᏮ.

Hu and Taylor [7] considered SLLN for arrays of row-wise independent random variables{Xni, 1≤i≤n, n≥1}.

Row-wise independence means that the random elements within each row are independent but no independence is assumed between rows.

In [3] Bozorgnia et al. obtained the Chung-type SLLN for arrays of row-wise independent random elements in a separable Banach space of Rademacher typep, 1< p≤2.

They proved the following result.

Theorem1. Let{Xni,1≤i≤n, n≥1}be an array of row-wise independent random elements in a separable Banach space of Rademacher typep,1< p≤2. Letϕ:R→R be a positive, even, and continuous function such that

ϕ

|x|

|x|^r , ϕ

|x|

|x|^r+p−1 as|x| , (6)

for some integerr≥2.

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Then the conditions

EXni=0, 1≤i≤n, n≥1, ∞

n=1

n i=1

EϕXni ϕ

an <∞, ∞ n=1



ⁿ

i=1

E Xni

an

^p 



pk

<∞, (7)

for some positive integerk, imply 1 an

n i=1

Xni →0 a.s.,n → ∞, (8)

where{an, n≥1}is a sequence of positive increasing real numbers such that

n→∞liman= ∞. (9)

This theorem generalizes Hu and Taylor’s result (cf. [7]) on the case ofᏮ-valued random variables{Xni,1≤i≤n, n≥1}taking value in a Banach space of Rademacher typep. Moreover, the assumptions of the functionϕhave some relationships with the geometric condition Rademacher typepof the Banach space.

Some results which consider the problem of equivalence between weak law of large numbers (WLLN) and SLLN for a sequence{Xn, n≥1}of independentᏮ-valued random variables can be found in Kuelbs and Zinn [10], de Acosta [4], Etemadi [5], Mikosch and Norvaiša [11,12], Wang et al. [14], and Kuczmaszewska and Szynal [9].

Now, we recall some deﬁnitions and a lemma which will be used in the paper.

Definition2. A double array{ani, i≥1, n≥1}of real numbers is said to be a Toeplitz array if lim_n→∞ani=0 for eachi≥1 and_∞

i=1|ani| ≤C for alln≥1, where C >0.

In further consideration, we need an extension of the concept of stochastic domina- tion by a random variable to an array ofᏮ-valued random variables.

An array{Xni, i≥1, n≥1}ofᏮ-valued random variables is stochastically dominated by the random elementXif there exists a constantD >0 such that

PXni> x

≤DP

DX> x

(10) for allx≥0,i≥1, andn≥1.

We also need some inequalities which will be very important in our consideration.

The following lemma presents one of them.

Lemma3(cf. Yurinskii [16]). LetX1,X2,...,Xnbe independentᏮ-valued random va- riables withEXi<∞,i=1,2,...,n. LetᏲ be aσ-field generated by(X1,X2,...,Xk), k=1,2,...,n, and letᏲ0= {Ω,∅}. Then for1≤k≤nandSn=_n

i=1Xi, ESnᏲk

−ESnᏲk−1≤Xk+EXk. (11) Theorem4. Let{Xni, 1≤i≤kn, n≥1}be an array of row-wise independentᏮ^- valued random variables withEXni=0for all1≤i≤kn,n≥1, and for some increasing

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sequence{kn, n≥1}of positive integers. Letϕni:R→R+andψni:R→R+be positive, even, and continuous functions, which for constantsαni≥1,0< βni≤2,Kni>0, and Mni>0,1≤i≤kn,n≥1, satisfy the following conditions:

x1≤x2 ⇒ϕnix1

x1^αⁿⁱ ≤Kniϕnix2

x2^αⁿⁱ , (12) x1≤x2 ⇒ x1^βⁿⁱ

ψnix1≤Mni

x2^βⁿⁱ

ψnix2. (13)

Suppose that for some array{ani, (1≤i≤kn, n≥1)}of nonzero reals andk≥1/2, ∞

n=1

E



^kⁿ

i=1

MniψniXni ψni

a⁻¹_ni





k

<∞, (14)

∞ n=1

kn

i=1

PXni≥cni

<∞, (15)

for some array{cni,1≤i≤kn, n≥1}of positive numbers such that ∞

n=1

n i=1

K_ni² ·ϕni

cniEϕniXni ϕ²_ni

a⁻¹_ni <∞. (16) Then

kn

i=1

aniXni P→0, n → ∞, (17)

if and only if

kn

i=1

aniXni →0 a.s.,n→ ∞. (18)

Proof. LetX_ni =XniI[Xni ≤ |a⁻_ni¹|]andX^∗_ni=X_ni −EX_ni . Now we introduce the following notation:

Sn=

kn

i=1

aniXni, S_n =

kn

i=1

aniX_ni . (19)

Note that using this notation, condition (12) on the Borel functionsϕni, and assumptions (15) and (16), we have

∞ n=1

P

Sn=S_n

= ∞ n=1

P





kn

i=1

aniXniIXni>a⁻¹_ni > ε





≤ ∞ n=1

kn

i=1

PXniIXni>a⁻_ni¹=0

= ∞ n=1

kn

i=1

PXni>a⁻¹_ni

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≤ ∞ n=1

kn

i=1

EIXni>a⁻¹_ni·IXni≥cni +

∞ n=1

kn

i=1

EIXni>a⁻_ni¹·IXni< cni

≤ ∞ n=1

kn

i=1

PXni≥cni +

∞ n=1

kn

i=1

E



 Xni^αⁿⁱ a⁻¹_ni^αⁿⁱ

2

IXni>a⁻_ni¹·IXni< cni





≤ ∞ n=1

kn

i=1

PXni≥cni +

∞ n=1

kn

i=1

K_ni² ·ϕni

cniEϕniXni ϕ_ni²

a⁻¹_ni <∞.

(20) Thus the two sequences{Sn, n≥1}and{S_n, n≥1}are equivalent.

Now we must prove that

ESn →0, n → ∞. (21)

First we will show that

S_n−ES_n ^P→0, n → ∞. (22) Using the Markov inequality, the Marcinkiewicz-Zygmunt inequality in its Banach space version (cf. de Acosta [4] or Berger [2]), and assumptions (12) and (14), for anyε >0, we get

PSn−ESn> ε

≤ε⁻²^kESn−ESn²^k

≤ε⁻²^kAkE



^kⁿ

i=1

aniX_ni ²



k

=ε⁻²^kAkE



^kⁿ

i=1

X_ni ² a⁻_ni¹2





k

=ε^−2kAkE



^kⁿ

i=1

X_ni ^βⁿⁱ

a⁻_ni¹^βⁿⁱ· X_ni ^2−βⁿⁱ a⁻_ni¹²^−βⁿⁱ





k

≤ε^−2kAkE



^kⁿ

i=1

MniψniX_ni ψni

a⁻¹_ni





k

×ε^−2kAkE



^kⁿ

i=1

MniψniXni ψni

a⁻¹_ni





k

=o(1).

(23)

Thus we conclude that (22) holds and, together with (17) and the equivalence between {Sn, n≥1}and{S_n, n≥1}, gives (21).

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Now, we will show thatS_n →0 a.s., asn→ ∞. By (21) it is enough to prove that Sn−ESn →0 a.s.,n→ ∞. (24) As before, using the Markov inequality, the Marcinkiewicz-Zygmunt inequality, condition (13), and assumption (14), we have

∞ n=1

PS_n−ES_n> ε

≤ε⁻²^k ∞ n=1

ES_n−ES_n²^k

≤ε⁻²^kAk

∞ n=1

E



^kⁿ

i=1

X_ni ² a⁻_ni¹2





k

=ε^−2kAk

∞ n=1

E



^kⁿ

i=1

X_ni ^βⁿⁱ

a⁻_ni¹^βⁿⁱ· X_ni ^2−βⁿⁱ a⁻_ni¹²^−βⁿⁱ





k

≤ε⁻²^kAk

∞ n=1

E



^kⁿ

i=1

MniψniXni ψni

a⁻_ni¹





k

<∞.

(25)

Hence, by the Borel-Cantelli lemma, we obtain (24), which, by the equivalence between {Sn, n≥1}and{S_n, n≥1}, completes the proof.

Note that if we put inTheorem 4ϕni=ϕandψni=ψ, whereϕ:R→R+,ψ:R→R+

are positive, even, and continuous functions such that ϕ

|x|

|x|^α , ψ

|x|

|x|^β as|x| , (26) for someα≥1 and 0< β≤2, we get the following result.

Corollary5. Let{Xni,1≤i≤kn, n≥1}be an array of row-wise independentᏮ- valued random variables withEXni=0for all1≤i≤kn,n≥1, and for any increasing sequence{kn, n≥1}of positive integers. Letϕ:R→R+ andψ:R→R+ be positive, even, and continuous functions satisfying (26) for someα≥1and0< β≤2.

Suppose that for some array{ani,1≤i≤kn, n≥1}of nonzero reals andk≥1/2, ∞

n=1

E



^kⁿ

i=1

ψXni ψ

a⁻_ni¹





k

<∞, ∞ n=1

kn

i=1

PXni≥cni

<∞, (27)

for some array{cni, (1≤i≤1, n≥1)}of positive numbers such that ∞

n=1 kn

i=1

ϕ

cniEϕXni ϕ²

a⁻_ni¹ <∞. (28)

Then (17) is equivalent to (18).

Puttingψ(x)= |x|^p,1< p≤β≤2, andcni= |a⁻¹_ni|, we obtain the following result for a separable Banach space of Rademacher typep.

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Corollary6. Let{Xni,1≤i≤kn, n≥1}be an array of row-wise independentᏮ^- valued random variables in a separable Banach space of Rademacher typep,1< p≤β≤ 2, withEXni=0for all1≤i≤kn,n≥1, and for some increasing sequence{kn, n≥1} of positive integers. Letϕ:R→Rbe a positive, even, and continuous function such that

ϕ

|x|

|x|^α as|x| , (29)

for someα≥1.

Then, for some array{ani, (1≤i≤kn, n≥1)}of nonzero reals and some integer k≥1, the conditions

∞ n=1



^kⁿ

i=1

EXni^p ani^−p





k

<∞, (30)

∞ n=1

kn

i=1

EϕXni ϕ

a⁻¹_ni <∞ (31)

imply (18).

Proof. PuttingMni=Kni=1 and using (20), we see that it is enough to show that Theorem 4holds for{Xni,1≤i≤kn, n≥1}andk≥2.

Indeed, we have ∞

n=1

E



^kⁿ

i=1

X_ni ^p a⁻¹_ni^p





k

≤ ∞ n=1

_∗ k s1,...,skn

E X_n1^p a⁻n1¹^p

s1

E X_n2^p a⁻n2¹^p

s2

···E X_nk _n^p a⁻_nk¹_n^p

s_kn

, (32)

where the sum_∗ _k

s1,...,s_kn

is over all choices of{s1,s2,...,skn},si∈ {0,1,2,...,k}, such that_k_n

i=1si=k. Choosensuﬃciently large so thatkn> k. Letm=m(s1,s2,...,skn)be a number ofsi=0. We see thatmtakes all the values from the set{1,2,...,k}. Changing the order in our sum, we can express the right-hand side of (32) in the following form:

∞ n=1

k m=1

1≤i_j≤kn, j=1,2,...,m ij=ik∀k=j

∗

k si1,...,sim

E



X_ni ₁^p a⁻_ni¹₁^p





si1

E



X_ni ₂^p a⁻_ni¹₂^p





si2

···E



X_ni_m^p a⁻_ni¹_m^p





s_im

≤ ∞ n=1











1≤i1<i2<···<ik≤kn

E



X_ni ₁^p a⁻¹_ni₁^p



E



X_ni ₂^p a⁻¹_ni₂^p



···E



X_ni _k^p a⁻¹_ni_k^p





(8)

+







k−1 m=1

1≤i_j≤kn, j=1,2,...,m i_j=i_k∀k=j

∗

k s1,...,sim





 L h=1s_ijh≥2

E



 X_ni

jh

^p a⁻_ni¹_jh^p





s_ijh











·





 N j=1

E

X_ni _hj^p a⁻¹_ni

hj

^p















 ,

(33)

whereL=number ofsi≥2,N=number ofsi=1, and{si1,...,sim}={si_jh,h=1,...,L}∪

{si_hj,si_hj=1,h=1,...,N},{si_jh,h=1,...,L}∩{si_hj,si_hj=1,h=1,...,N}=∅.

But



 X_ni _j^p a⁻¹_ni_j^p





s_ij

≤

X_ni _j^p a⁻¹_ni_j^p, E

X_ni _j^p a⁻¹_ni_j^p ≤

kn

i=1

EX_ni _j^p

a⁻¹_ni_j^p for 1≤ij≤kn, (34) so the right-hand side of (33) can be estimated as follows:

C ∞ n=1









^kⁿ

i=1

EX_ni ^p a⁻_ni¹^p





k

+



^kⁿ

i=1





L

·



^kⁿ

i=1





M





≤C ∞ n=1



^kⁿ

i=1

EX_ni^p a⁻_ni¹^p





k

<∞.

(35)

Therefore, assumption (14) ofTheorem 4holds. Moreover, by (31), we get (15) and (16) ofTheorem 4. We also note that ifᏮis a Banach space of Rademacher typep, 1< p≤2, we have the following estimation:

PSn≥ε

≤ε^−pESn^p≤ε

kn

i=1

EaniXni^p≤ε^−p

kn

i=1

EXni^p

ani^−p =o(1). (36)

This fact, together with (20), completes the proof.

Now we present the result which gives the suﬃcient conditions for the equivalence of (17) and (18) in the Chung-Teicher terms (cf. Teicher [13]).

Theorem7. Let{Xni, 1≤i≤kn, n≥1}be an array of row-wise independentᏮ- valued random variables withEXni=0for all1≤i≤kn,n≥1, and for some increas- ing sequence{kn, n≥1}of positive integers. Letϕni:R→R+ be positive, even, and

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continuous functions which, for constantsαni≥1, 0< βni≤2,Kni>0, andMni>0, n≥1,i≥1, satisfy (12) and

x1≤x2 ⇒ x1^βⁿⁱ

ϕnix1≤Mni

x2^βⁿⁱ

ϕnix2. (37)

Suppose that for some array{ani, (1≤i≤kn, n≥1)}of nonzero reals, ∞

n=1 kn

i=2

MniEϕniXni Pϕni

a⁻¹_ni

i−1

j=1

MnjEϕnjXnj ϕnj

a⁻¹_nj <∞, (38)

kn

i=1

MniEϕniXni ϕni

a⁻_ni¹ =o(1), (39)

kn

i=1

KniEϕniXni ϕni

a⁻¹_ni =o(1), (40)

∞ n=1

kn

i=1

PXni≥cni

<∞, (41)

for some array{cni, i≥1, n≥1}of positive numbers such that ∞

n=1 kn

i=1

M_ni² ·ϕni

a⁻¹_ni <∞, (42) ∞

n=1 kn

i=1

K_ni² ·ϕni

a⁻_ni¹ <∞. (43) Then (17) is equivalent to (18).

Proof. LetX_ni =XniI[Xni ≤ |a⁻¹_ni|],X_ni^∗ =X_ni−EX_ni ,S_n =_k_n

i=1aniX_ni, andS_n^∗= _k_n

i=1aniX_ni^∗. By (20), we state that{Sn, n≥1}and{Sn, n≥1}are equivalent.

Moreover, we have by (40)

kn

i=1

aniEXniIXni≤a⁻¹_ni ≤

kn

i=1

ani·EXniIXni≤a⁻¹_ni

=

kn

i=1

ani·EXniIXni>a⁻_ni¹

≤

kn

i=1

EXniIXni>a⁻_ni¹ a⁻¹_ni

≤

kn

i=1

EXni^αⁿⁱIXni>a⁻¹_ni a⁻¹_ni^αⁿⁱ

≤

kn

i=1

Kni·EϕniXni ϕni

a⁻¹_ni =o(1).

(44)

(10)

452 ANNA KUCZMASZEWSKA Now we deﬁne

Yni=ES_n^∗Ᏺni

−ES_n^∗Ᏺni−1

, (45)

whereᏲni=σ (X_n^∗1,X_n^∗2,...,X_ni^∗)andᏲn0= {∅,Ω}. Then we have Sn^∗−ESn^∗=

kn

i=1

Yni (46)

and we note that{Yni,1≤i≤kn}is a sequence of martingale diﬀerences for a ﬁxedn. Now we are going to prove that

ES_n^∗ →0, n → ∞. (47)

First we will show that

S_n^∗−ES_n^∗ ^P→0, n → ∞. (48)

Using Chebyshev’s inequality,Lemma 3, and assumption (39), we get, for anyε >0, PS_n^∗−ES_n^∗> ε

≤ε⁻²ES_n^∗−ES_n^∗²=ε⁻²E



^kⁿ

i=1

Yni





2

=ε⁻²

kn

i=1

E Y_ni²

≤ε⁻²

kn

i=1

EaniX_ni^∗+EaniX_ni^∗²≤8ε⁻²

kn

i=1

a²_niEX_ni ²

=8ε⁻²

kn

i=1

EX_ni ^βⁿⁱ

a⁻_ni¹^βⁿⁱ·X_ni ^2−βⁿⁱ a⁻_ni¹²^−βⁿⁱ ≤8ε⁻²

kn

i=1

MniEϕniX_ni ϕni

ani =o(1).

(49)

Thus, we conclude that (48) holds and, together with (17), (20), and (44), gives (47).

Now we want to show thatS_n^∗ →0 a.s., asn→ ∞. By (47) it is enough to prove that

Sn^∗−ESn^∗ →0 a.s.,n→ ∞. (50)

Taking into account the identity

S_n^∗−ES_n^∗²=

kn

i=1

Y_ni²+2

kn

i=2

Yni i−1

j=1

Ynj (51)

and using the notation

Zni=Y_ni²IXni< cni

−E

Y_ni²IXni< cniᏲni−1

, 1≤i≤kn, (52)

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we have, by Chebyshev’s inequality,Lemma 3, and assumption (42), ∞

n=1

P





kn

i=1

Zni

> ε



≤ε⁻² ∞ n=1

E



^kⁿ

i=1

Zni





2

=ε⁻² ∞ n=1

kn

i=1

E Z_ni²

≤C·ε⁻² ∞ n=1

kn

i=1

E

Y_ni⁴IXni< cni

≤C·ε⁻² ∞ n=1

kn

i=1

EaniX_ni ⁴IXni< cni

≤C·ε⁻² ∞ n=1

kn

i=1

E



X_ni ²^βⁿⁱ

a⁻¹_ni^2βⁿⁱ·X_ni ⁴⁻²^βⁿⁱ

a⁻¹_ni^4−2βⁿⁱIXni< cni



≤C·ε⁻² ∞ n=1

kn

i=1

Mni·ϕni cni

·EϕniXni ϕ²_ni

a⁻¹_ni <∞.

(53)

Hence, by the Borel-Cantelli Lemma, we obtain

kn

i=1

Y_ni²IXni< cni

−

kn

i=1

E

Y_ni²IXni< cniᏲni−1

→0 a.s.,n → ∞. (54)

UsingLemma 3, we can note by assumption (39) that

kn

i=1

E

Y_ni²IXni< cniᏲni−1

≤8

kn

i=1

EaniX_ni ²

≤

kn

i=1

E



X_ni ^βⁿⁱ

a⁻¹_ni^βⁿⁱ·X_ni ²^−βⁿⁱ a⁻¹_ni^2−βⁿⁱ





≤

kn

i=1

Mni·EϕniXni ϕni

a⁻¹_ni =o(1),

(55)

which, together with (54), allows us to state that

kn

i=1

Y_ni²IXni< cni

→0 a.s.,n → ∞. (56)

To prove that

kn

i=1

Y_ni² →0 a.s.,n → ∞, (57)

we only need to show that

kn

i=1

Y_ni²IXni≥cni

→0 a.s.,n → ∞. (58)

(12)

454 ANNA KUCZMASZEWSKA Indeed, by (41) andLemma 3, we have

∞ n=1

P





kn

i=1

Y_ni²IXni≥cni ≥ε





≤ε⁻¹ ∞ n=1

E

kn

i=1

Y_ni²IXni≥cni ≤C

∞ n=1

kn

i=1

EaniX_ni ²IXni≥cni

≤C ∞ n=1

kn

i=1

EIXni≥cni

≤ε⁻¹ ∞ n=1

kn

i=1

PXni≥cni

<∞,

(59)

and, by the Borel-Cantelli Lemma, we get (57). To close this proof, we must show that

kn

i=2

Yni i−1 j=1

Ynj →0 a.s.,n→ ∞. (60)

Using the fact that{Yni_i−1

j=1Ynj, 2≤i≤kn}is a sequence of martingale diﬀerences for eachn, we have, by Chebyshev’s inequality,Lemma 3, and assumption (38),

∞ n=1

P





kn

i=2

Yni i−1

j=1

Ynj

> ε





≤ε⁻² ∞ n=1

kn

i=2

E



aniX_ni^∗+EaniX_ni^∗²·



ⁱ⁻¹

j=1

Ynj





2



≤ε⁻² ∞ n=1

kn

i=2

EaniX_ni^∗+EaniX^∗_ni²·

i−1

j=1

E Y_nj²

≤C ∞ n=1

kn

i=2

EaniX_ni²ⁱ⁻¹

j=1

EanjX_nj ²C ∞ n=1

kn

i=2

MniEϕniXni ϕni

a⁻¹_ni

×

i−1 j=1

MnjEϕnjXnj ϕnj

a⁻_nj¹ <∞,

(61)

which, together with the Borel-Cantelli Lemma, implies (60).

Thus, we have proved that

kn

i=1

ani

X_ni −EX_ni

→0 a.s.,n→ ∞. (62)

But{_k_n

i=1aniX_ni , n≥1}and{_k_n

i=1aniXni, n≥1}are equivalent and (44) holds, so we get (18).

Now we consider an array{Xni, i≥1, n≥1}of independent random elements which are stochastically dominated by a random elementXin the sense of (10).

Corollary 8. Let{Xni, 1≤i≤kn, n≥1}be an array of row-wise independent Ꮾ-valued random variables withEXni=0for all1≤i≤kn,n≥1, and some increasing