Lobachevskii Journal of Mathematics Vol. 10, 2002, 17 – 26 \begin{matrix}c \\ ◯ \end{matrix}T.-C. Hu, M. Ordóñez Cabrera, S. H. Sung, A.I. Volodin

T.-C. Hu, M. Ordóñez Cabrera, S. H. Sung, A.I. Volodin

COMPLETE CONVERGENCE OF WEIGHTED SUMS IN BANACH SPACES AND THE BOOTSTRAP MEAN

(submitted by D. Kh. Mushtari)

Abstract. Let {X_{ni}, 1 ≤ i ≤ k_{n}, n ≥ 1} be an array of rowwise independent random elements taking values in a real separable Banach space, and {a_{ni}, 1 ≤ i ≤ k_{n}, n ≥ 1} an array of constants. Under some conditions of Chung [7] and Hu and Taylor [10] types for the arrays, and using a theorem of Hu et al. [9], the equivalence amongst various

kinds of convergence of ∑_{i=1}^{k_{n}} a_{ni}X_{ni} to zero is obtained. It leads to an

unified vision of recent results in the literature. The authors use the main result in the paper in order to obtain the strong consistency of the bootstrapped mean of random elements in a Banach space from its weak consistency.

1. INTRODUCTION

Hsu and Robbins [8] introduced the concept of complete convergence of a sequence {X_{n}, n ≥ 1} of random variables. {X_{n}, n ≥ 1} is said to converge completely to a constant c if for each ϵ > 0

∑_{n=1}^{∞} P [|X_{n} − c| > ϵ] < ∞.

This concept is extended to a sequence {X_{n}, n ≥ 1} of random elements taking values in a real separable Banach space (B, ||.||). It is said that {X_{n}} converges completely to zero if for each ϵ > 0

∑_{n=1}^{∞} P [||X_{n}|| > ϵ] < ∞.

18 T.-C. HU, M. ORDÓÑEZ CABRERA, S. H. SUNG, A.I. VOLODIN

Classical Chung’s [7] strong law of large numbers (SLLN) states that if {X_{n}, n ≥ 1} is a sequence of independent random variables with EX_{n} = 0 for all n ≥ 1 and ∑_{n=1}^{∞} \frac{Eψ(|X_{n}|)}{ψ(n)} < ∞ when ψ is a positive, even and continuous

1991 Mathematical Subject Classification. Primary 60B12, 62E20, 62G05; Secondary 60B11.

Key words and phrases. Random elements, Banach spaces, weighted sums, rowwise independence, complete convergence, bootstrap mean.

function such that

\frac{ψ(|t|)}{|t|} ↑ and \frac{ψ(|t|)}{|t|^{2}} ↓ as |t| ↑  (1.1)

then \frac{1}{n} ∑_{i=1}^{n} X_{i} ⟶ 0 almost surely.

Hu and Taylor [10] proved a Chung type SLLN for an array of rowwise independent random variables {X_{ni}, 1 ≤ i ≤ n, n ≥ 1} weighted by a sequence {a_{n}, n ≥ 1} of real numbers with 0 < a_{n} ↑ ∞. One of the required conditions

by the authors in order to prove that \frac{1}{a_{n}} ∑_{i=1}^{n} X_{ni} ⟶ 0 almost surely, is that

∑_{n=1}^{∞} ∑_{i=1}^{n} \frac{Eψ(|X_{ni}|)}{ψ(a_{n})} < ∞,

when ψ is a positive, even and continuous function such that

\frac{ψ(|t|)}{|t|^{p}} ↑ and \frac{ψ(|t|)}{|t|^{p+1}} ↓ as |t| ↑,  (1.2)

for some integer p ≥ 2.

Many classical theorems hold for B-valued random elements under the assumption that the weak law of large numbers (WLLN) holds (see Kuelbs and Zinn [12], de Acosta [1], Choi and Sung [5, 6], Wang et al. [16], Kuczmaszeva and Szynal [11] and Sung [14]).

Sung [15] obtained a Hu and Taylor’s [10] result in a general Banach space under the assumption that WLLN holds, using a positive and even function ψ verifying the following condition, which is weaker than (1.2):

\frac{ψ(|t|)}{|t|} ↑ and \frac{ψ(|t|)}{|t|^{p}} ↓ as |t| ↑,  (1.3)

for some p ≥ 1.

Ordóñez Cabrera and Sung [13] extend Sung’s [15] result for a weighted

sum S_{n} = ∑_{i=1}^{k_{n}} a_{ni}X_{ni} of rowwise independent B-valued random elements, where

{k_{n}, n ≥ 1} is a sequence of positive integers, and {a_{ni}, 1 ≤ i ≤ k_{n}, n ≥ 1} is an array of real constants. They use a function ψ : (0, +∞) ⟶ (0, +∞) verifying that, for some p ≥ 1, there exist constants C, D > 0 such that

u ≥ v ⟹ C \frac{u}{v} ≤ \frac{ψ(u)}{ψ(v)} ≤ D\frac{u^{p}}{v^{p}}  (1.4)

This condition is weaker than (1.3).

In the main result of this paper, we obtain the equivalence between the convergence to zero in L^{1}, completely, almost surely and in probabillity, for

a weighted sum S_{n} = ∑_{i=1}^{k_{n}} a_{ni}X_{ni} of rowwise independent B-valued random

elements, using a function ψ verifying a condition weaker than (1.4), and this allows us to give an unified overall view of recent results in the literature that we mentioned before.

The proof of our main result is based on the following theorem, proved in Hu et al. [9], Theorem 3.2 (we took c_{n} = 1 for all n ≥ 1 and weights are built into random elements).

Theorem 1.1. Let {X_{ni}, 1 ≤ i ≤ k_{n}, n ≥ 1} be an array of rowwise independent random elements in a real separable Banach space such that

∑_{n=1}^{∞} ∑_{i=1}^{k_{n}} P {||X_{ni}|| > ϵ} < ∞ for all ϵ > 0,

there exist s > 0 such that for some 0 < r ≤ 2

∑_{n=1}^{∞} (∑_{i=1}^{k_{n}} E||X_{ni}||^{r})^{s} < ∞,

and

∑_{i=1}^{k_{n}} X_{ni} →^{P} 0.

Then

∑_{n=1}^{∞} P {|| ∑_{i=1}^{k_{n}} X_{ni}|| > ϵ} < ∞ for all ϵ > 0.

To prove the main result, we also will need the following lemma in de Acosta [1] (cf. also Berger [3]):

Lemma 1.1. Let {X_{nk}, 1 ≤ k ≤ k_{n}, n ≥ 1} be an array of rowwise independent random elements. Then for every p ≥ 1/2, there is a positive constant A_{p} depending only on p such that for all n ≥ 1

E||| ∑_{i=1}^{k_{n}} X_{ni}|| − E|| ∑_{i=1}^{k_{n}} X_{ni}|||^{2p} ≤ A_{p}E (∑_{k=1}^{k_{n}} ||X_{nk}||^{2})^{p} .

In the last section, we apply the results of previous section to the field of bootstrapped means for random elements, after making a brief summary of the bootstrap procedure. Our result is in the scope of the previous ones of Bozorgnia et al. [4] and Hu and Taylor [10]. More specifically, we obtain strong consistency (concerning to an array of real constants {a_{ni}, 1 ≤ i ≤ k_{n}, n ≥ 1}) for the bootstrapped mean, assuming the corresponding weak consistency (concerning the array). We impose neither conditions about the marginal or joint distributions of the random elements forming the sample nor geometric conditions on the Banach space where the random elements take values.

2. PRELIMINARIES

Definition 2.1. Let ψ : (0, +∞) ⟶ (0, +∞) be a function such that there exists a constant C > 0 such that

u ≥ v ⟹ ψ(u) ≥ Cψ(v)  (2.1)

and for any ϵ > 0

sup_{u>0} \frac{ψ(u)}{ψ(ϵu)} < ∞  (2.2)

By putting u = v (or by using a continuity argument, if ψ is continuous), it is clear that 0 < C ≤ 1.

Conditions (2.1) and (2.2) are weaker than conditions (1.3) and (1.4). Consequently, the family of functions verifying (2.1) and (2.2) is wider than the

family of functions verifying \frac{ψ(x)}{x} ↑ and \frac{ψ(x)}{x^{p}} ↓. The following Lemma shows a sufficient condition so that ψ should verify (2.1) and (2.2). Lemma 2.1. Let ψ : (0, +∞) ⟶ (0, +∞) be a function such that Ax ≤ ψ(x) ≤ Bx for all x ∈ (0, +∞), for some constants A, B > 0. Then:

\frac{A}{B} \frac{u}{v} ≤ \frac{ψ(u)}{ψ(v)} ≤ \frac{B}{A} \frac{u}{v}

for all u, v > 0.

Throughout this paper, unless otherwise specified, ψ will be a function satisfying (2.1) and (2.2).

3. MAIN RESULT

Theorem 3.1. Let k_{n} → ∞ be a sequence of positive integers. Let {X_{ni}, 1 ≤ i ≤ k_{n}, n ≥ 1} be an array of rowwise independent B-valued random elements. Assume that

∑_{n=1}^{∞} ∑_{i=1}^{k_{n}} P {||X_{ni}|| > ϵ} < ∞ for all ϵ > 0,  (3.1)

and

∑_{n=1}^{∞} (∑_{i=1}^{k_{n}} E||X_{ni}||^{r})^{s} < ∞  (3.2)

for some s > 0 and some 1 < r ≤ 2. Then the following statements are equivalent:

(i) ∑_{i=1}^{k_{n}} X_{ni} ⟶ 0 in L^{1}.

(ii) ∑_{i=1}^{k_{n}} X_{ni} ⟶ 0 completely.

(iii) ∑_{i=1}^{k_{n}} X_{ni} ⟶ 0 almost surely.

(iv) ∑_{i=1}^{k_{n}} X_{ni} ⟶ 0 in probability.

Proof.(ii) ⟹ (iii), (iii) ⟹ (iv) and (i) ⟹ (iv) are immediate. (iv) ⟹ (ii) is proved in Theorem 1.1.

(iv) ⟹ (i). Assume that (iv) holds. From (3.2) and Lemma 1.1 with p = r/2:

E ||| ∑_{i=1}^{k_{n}} X_{ni}|| − E|| ∑_{i=1}^{k_{n}} X_{ni}|||^{r} ≤ A_{r/2}E(∑_{i=1}^{k_{n}} ||X_{ni}||^{2})^{r/2}

≤ A_{r/2} ∑_{i=1}^{k_{n}} E||X_{ni}||^{r} ⟶ 0

and so

|| ∑_{i=1}^{k_{n}} X_{ni}|| − E|| ∑_{i=1}^{k_{n}} X_{ni}|| ⟶ 0 in probability.

Consequently, E|| ∑_{i=1}^{k_{n}} X_{ni}|| ⟶ 0, and so (i) holds. □

Theorem 3.2. Let k_{n} → ∞ be a sequence of positive integers. Let {X_{ni}, 1 ≤ i ≤ k_{n}, n ≥ 1} be an array of rowwise independent B-valued random elements, and {a_{ni}, 1 ≤ i ≤ k_{n}, n ≥ 1} an array of constants. Assume that

∑_{n=1}^{∞} ∑_{i=1}^{k_{n}} \frac{Eψ(||X_{ni}||)}{ψ(|a_{ni}|^{−1})} < ∞  (3.3)

and

∑_{n=1}^{∞} (∑_{i=1}^{k_{n}} |a_{ni}|^{r}E||X_{ni}||^{r})^{s} < ∞  (3.4)

for some 1 ≤ r ≤ 2 and some s > 0. Then the following statements are equivalent:

(i) ∑_{i=1}^{k_{n}} a_{ni}X_{ni} ⟶ 0 in L^{1}.

(ii) ∑_{i=1}^{k_{n}} a_{ni}X_{ni} ⟶ 0 completely.

(iii) ∑_{i=1}^{k_{n}} a_{ni}X_{ni} ⟶ 0 almost surely.

(iv) ∑_{i=1}^{k_{n}} a_{ni}X_{ni} ⟶ 0 in probability.

Proof. Consider a_{ni}X_{ni} insterd of X_{ni} in Theorem 3.1. The only thing we need to prove is that (3.3) ⟹ (3.1).

For each ϵ > 0 by (2.1), Markov inequality and (2.2):

∑_{n=1}^{∞} ∑_{i=1}^{k_{n}} P [||a_{ni}X_{ni}|| > ϵ]

≤ ∑_{n=1}^{∞} ∑_{i=1}^{k_{n}} P [ψ(||X_{ni}||) > Cψ(ϵ|a_{ni}|^{−1})] ≤ \frac{1}{C} ∑_{n=1}^{∞} ∑_{i=1}^{k_{n}} \frac{E[ψ(||X_{ni}||)]}{ψ(ϵ|a_{ni}|^{−1})}

≤ C′ ∑_{n=1}^{∞} ∑_{i=1}^{k_{n}} \frac{E[ψ(||X_{ni}||)]}{ψ(|a_{ni}|^{−1})}.□

Remark 3.1. An analysis of the proof of the theorem 3.2 shows that the condition (2.2) can be simplified as

sup_{n,i} \frac{ψ(|a_{ni}|^{−1})}{ψ(ϵ|a_{ni}|^{−1})} < ∞

for any ϵ > 0

We obtain the complete convergence of weighted sums taking values in a Banach space of type r (1 ≤ r ≤ 2) as a corollary of this theorem.

Recall that a separable Banach space B is of type r, 1 ≤ r ≤ 2, if, and only if, there exists a constant C_{r} such that

E|| ∑_{i=1}^{n} X_{i}||^{r} ≤ C_{r} ∑_{i=1}^{n} E||X_{i}||^{r}

for all independent B-valued random elements X_{1}, ..., X_{n} with mean zero and finite r-th moments.

Corollary 3.1. Let k_{n} → ∞ be a sequence of positive integers. Let {X_{ni}, 1 ≤ i ≤ k_{n}, n ≥ 1} be an array of rowwise independent B-valued random elements, and {a_{ni}, 1 ≤ i ≤ k_{n}, n ≥ 1} an array of constants. Assume that EX_{ni} = 0 for all 1 ≤ i ≤ k_{n} and n ≥ 1, and that B is of type r (1 ≤ r ≤ 2). Assume (3.1) and (3.4) hold for this r and some s > 0. Then,

∑_{i=1}^{k_{n}} a_{ni}X_{ni} ⟶ 0 completely.

Proof. By Theorem 3.2, it is enough to prove that ∑_{i=1}^{k_{n}} a_{ni}X_{ni} ⟶ 0 in L^{1}.

Since B is of type r and E(a_{ni}X_{ni}) = 0, we have

E|| ∑_{i=1}^{k_{n}} a_{ni}X_{ni}||^{r} ≤ C_{r} ∑_{i=1}^{k_{n}} |a_{ni}|^{r}E||X_{ni}||^{r} ⟶ 0

as a consequence of (3.4).

Convergence in L^{r} implies convergence in L^{1}, so Theorem 3.2 implies that

∑_{i=1}^{k_{n}} a_{ni}X_{ni} ⟶ 0 completely.□

Remark 3.2. If C = 1, k_{n} = n and a_{ni} = \frac{1}{a_{n}} , 1 ≤ i ≤ k_{n}, n ≥ 1, then The-

orem 3.2 and Corollary 3.1 become Theorem 2.2, Theorem 2.3 and Corollary 2.4 in Sung [15].

By using functions ψ satisfyimg the hypothesis of Lemma 2.1, we can weaken slightly the conditions (3.2) and (3.4) in Theorem 3.2.

Theorem 3.3. Let k_{n} → ∞ be a sequence of positive integers. Let {X_{ni}, 1 ≤ i ≤ k_{n}, n ≥ 1} be an array of rowwise independent B-valued random elements, and {a_{ni}, 1 ≤ i ≤ k_{n}, n ≥ 1} an array of constants. Let ψ : (0, +∞) ⟶ (0, +∞) be a function such that Ax ≤ ψ(x) ≤ Bx for all x ∈ (0, +∞), for some constants A, B > 0. Assume that

∑_{n=1}^{∞} ∑_{i=1}^{k_{n}} \frac{Eψ(||X_{ni}||)}{ψ(|a_{ni}|^{−1})} < ∞.

Then the following statements are equivalent:

(i) ∑_{i=1}^{k_{n}} a_{ni}X_{ni} ⟶ 0 in L^{1}.

(ii) ∑_{i=1}^{k_{n}} a_{ni}X_{ni} ⟶ 0 completely.

(iii) ∑_{i=1}^{k_{n}} a_{ni}X_{ni} ⟶ 0 almost surely.

(iv) ∑_{i=1}^{k_{n}} a_{ni}X_{ni} ⟶ 0 in probability.

Proof. Mention that in this case

\frac{A}{B} ||a_{ni}X_{ni}|| ≤ \frac{ψ(||X_{ni}||)}{ψ(|a_{ni}|^{−1})}.

For (i) ⟹ (ii) we refer to Corollary 4.7 in [9]. (iv) ⟹ (i). E|| ∑_{i=1}^{k_{n}} a_{ni}X_{ni}|| ≤

∑_{i=1}^{k_{n}} |a_{ni}|E||X_{ni}|| ⟶ 0, and (i) holds. □

Similarly, it is easy to check the following result of complete convergence in a Banach space of type 1 ≤ r ≤ 2.

Corollary 3.2. Let k_{n} → ∞ be a sequence of positive integers. Let {X_{ni}, 1 ≤ i ≤ k_{n}, n ≥ 1} be an array of rowwise independent B-valued random elements, and {a_{ni}, 1 ≤ i ≤ k_{n}, n ≥ 1} an array of constants. Let ψ : (0, +∞) ⟶ (0, +∞) be a function such that Ax ≤ ψ(x) ≤ Bx for all x ∈ (0, +∞), for some constants A, B > 0. Assume that EX_{ni} = 0 for all 1 ≤ i ≤ k_{n} and n ≥ 1, and B is of type r (1 ≤ r ≤ 2). Assume (3.2) holds.Then

∑_{i=1}^{k_{n}} a_{ni}X_{ni} ⟶ 0 completely.

4. APPLICATION TO THE STUDY OF THE CONSISTENCY OF THE BOOTSTRAPPED MEAN

We now outline the bootstrap procedure. Let {X_{n}; n ≥ 1} be a sequence of (not necessarily independent or identically distributed) random elements defined on some complete probability space (Ω, F, P) which take values in a

real separable Banach space. For ω ∈ Ω and n ≥ 1, let P_{n}(ω) = \frac{1}{n} ∑_{i=1}^{n} δ_{X_{i}(ω)}

denote the empirical measure. For n ≥ 1, let {X̂_{n,j}^{ω}; 1 ≤ j ≤ k_{n}} be i.i.d. random elements with law P_{n}(ω) where k_{n} is a positive integer. Let X̄_{n}(ω) denote the sample mean of {X_{i}(ω); 1 ≤ i ≤ n}, n ≥ 1, that is, X̄_{n}(ω) =

\frac{1}{n} ∑_{i=1}^{n} X_{i}(ω) .

In order to prove Theorem 4.1, we need the following lemma:

Lemma 4.1. If s > 0 then for almost every ω ∈ Ω

E||X̂_{n,1}^{ω} − X̄_{n}(ω)||^{s} ≤ A_{s} [\frac{1}{n} ∑_{i=1}^{n} ||X_{i}(ω)||^{s} + ||X̄_{n}(ω)||^{s}] ,

where A_{s} = 2^{s−1} for s ≥ 1 and A_{s} = 1 for 0 < s < 1.

Proof. For almost every ω ∈ Ω ,

E||X̂_{n,j}^{ω} − X̄_{n}(ω)||^{s} = \frac{1}{n} ∑_{i=1}^{n} ||X_{i}(ω) − X̄_{n}(ω)||^{s}

≤ A_{s} [\frac{1}{n} ∑_{i=1}^{n} (||X_{i}(ω)||^{s} + ||X̄_{n}(ω)||^{s})] ,

by the c_{r}-inequalities. □

Theorem 4.1. Let {X_{n}, n ≥ 1} be a sequence of random elements taking values in a real separable Banach space. Let k_{n} → ∞ be a sequence of positive integers, and let {a_{nk}, 1 ≤ k ≤ k_{n}, n ≥ 1} be an array of real constants. Let

1 ≤ r ≤ 2. Denote h_{n} = ∑_{k=1}^{k_{n}} |a_{nk}|^{r}, for every n ≥ 1. Let {b_{n}, n ≥ 1} and

{d_{n}, n ≥ 1} be sequences of positive constants. Suppose that:

(1) sup_{n≥1} \frac{1}{d_{n}} ||X̄_{n}|| < ∞ a.e. and sup_{n≥1} \frac{1}{b_{n}} ∑_{i=1}^{n} ||X_{i}||^{r} < ∞ a.e.,

(2) ∑_{n=1}^{∞} \frac{h_{n}b_{n}}{n} < ∞ and ∑_{n=1}^{∞} h_{n}d_{n}^{r} < ∞,

(3) The bootstrapped mean is weakly consistent concerning to {a_{nk}, 1 ≤ k ≤ k_{n}, n ≥ 1}, that is, for almost every ω ∈ Ω

|| ∑_{k=1}^{k_{n}} a_{nk} (X̂_{n,k}^{ω} − X̄_{n}(ω)) || →^{P} 0.

Then the bootstrapped mean is strongly consistent concerning to {a_{nk}, 1 ≤ k ≤ k_{n}, n ≥ 1}, that is, for almost every ω ∈ Ω and all ϵ > 0,

∑_{n=1}^{∞} P {|| ∑_{k=1}^{k_{n}} a_{nk} (X̂_{n,k}^{ω} − X̄_{n}(ω)) || > ϵ} < ∞.

Proof. If we consider the function ψ(t) = t^{r}, the condition (3.3) in Theorem

3.2 becomes ∑_{n=1}^{∞} ∑_{k=1}^{k_{n}} |a_{nk}|^{r}E||X_{nk}||^{r} < ∞, which implies condition (3.4) with

s = 1. Therefore, we need only check condition (3.3) for the array {Z_{nk} = X̂_{n,k}^{ω} − X̄_{n}(ω), 1 ≤ k ≤ k_{n}, n ≥ 1} with ψ(t) = t^{r}.

An application of the above lemma yields for almost every ω ∈ Ω:

∑_{n=1}^{∞} ∑_{k=1}^{k_{n}} |a_{nk}|^{r}E||Z_{nk}||^{r} = ∑_{n=1}^{∞} ∑_{k=1}^{k_{n}} |a_{nk}|^{r}E||X̂_{n,k}^{ω} − X̄_{n}(ω)||^{r}

≤ 2^{r−1} ∑_{n=1}^{∞} ∑_{k=1}^{k_{n}} |a_{nk}|^{r} [\frac{1}{n} ∑_{i=1}^{n} ||X_{i}(ω)||^{r} + ||X̄_{n}(ω)||^{r}]

= 2^{r−1} (∑_{n=1}^{∞}(\frac{1}{n} ∑_{i=1}^{n} ||X_{i}(ω)||^{r}) ∑_{k=1}^{k_{n}} |a_{nk}|^{r} + ∑_{n=1}^{∞} ||X̄_{n}(ω)||^{r} ∑_{k=1}^{k_{n}} |a_{nk}|^{r})

= 2^{r−1} (∑_{n=1}^{∞} \frac{h_{n}}{n} ∑_{i=1}^{n} ||X_{i}(ω)||^{r} + ∑_{n=1}^{∞} h_{n}||X̄_{n}(ω)||^{r})

COMPLETE CONVERGENCE OF WEIGHTED SUMS 25

= 2^{r−1} (∑_{n=1}^{∞} \frac{h_{n}b_{n}}{n} [\frac{1}{b_{n}} ∑_{i=1}^{n} ||X_{i}(ω)||^{r}] + ∑_{n=1}^{∞} h_{n}d_{n}^{r} [\frac{1}{d_{n}} ||X̄_{n}(ω)||]^{r}) < ∞

in view of (1) and (2).

By applying Theorem 3.2, we have that ∑_{k=1}^{k_{n}} a_{nk}Z_{nk} ⟶ 0 completely, that

is, the bootstrapped mean is strongly consistent concerning to {a_{nk}, 1 ≤ k ≤ k_{n}, n ≥ 1}. □

Remark 4.1. In the particular case a_{nk} = \frac{1}{a_{n}} for all 1 ≤ k ≤ k_{n} and every

n ≥ 1, where {a_{n}, n ≥ 1} is a sequence of positive constants, we have Theorem 3 of Ahmed, Hu and Volodin [2], with q = r ∈ [1, 2] and c_{n} = 1 for every n ≥ 1.

ACKNOWLEDGMENTS

The first author has been partially supported by DGICYT grant PB-96-1338-C02-01 and Junta de Andalucia. Part of the research of A.Volodin was conducted at the Department of Mathematical Analysis, University of Sevilla and he expresses his appreciation for the use of their facilities.

References

[1] de Acosta, A., Inequalities for B-valued random vectors with applications to the strong law of large numbers, Ann. Probab. 9 (1981), 157-161

[2] Ahmed, S.E., Hu, T.C. and Volodin, A.I., On the rate of convergence of botstrapped means in a Banach space, Internat. J. Math. & Math. Sci. 25 (2001), 629-635.

[3] Berger, E., Majorization, exponential inequalities and almost sure behavior of vectorvalued random variables, Ann. Probab. 19 (1991), 1206-1226.

[4] Bozorgnia, A., Patterson, R.F. and Taylor, R.L., Chung type strong laws for arrays of random elements and bootstrapping, Stochastic Anal. Appl. 15 (1997), 651-669.

[5] Choi, B.D. and Sung, S.H., On Chung’s strong law of large numbers in general Banach spaces, Bull. Austral. Math. Soc. 37 (1988), 93-100.

[6] Choi, B.D. and Sung, S.H., On Teicher’s strong law of large numbers in general Banach spaces, Probab. Math. Statist. 10 (1989), 137-142.

[7] Chung, K.L., Note on some strong laws of large numbers, Amer. J. Math. 69 (1947), 189-192.

[8] Hsu, P.L. and Robbins, H., Complete convergence and the law of large numbers, Proc. Nat. Acad. Sci. U.S.A. 33 (1947), 25-31.

[9] Hu, T.-C., Rosalsky, A., Szynal, D., Volodin, A., On complete convergence for arrays of rowwise independent random elements in Banach spaces, Stochastic Anal. Appl., 17, (1999), 963-992.

[10] Hu, T.C. and Taylor, R.L., On the strong law for arrays and for the bootstrap mean and variance Internat. J. Math. & Math. Sci. 20 (1997), 375-382.

[11] Kuczmaszewska, A. and Szynal, D., On complete convergence in a Banach space, Internat. J. Math. & Math. Sci. 17 (1994), 1-14.

[12] Kuelbs, J. and Zinn, J., Some stability results for vector valued random variables, Ann. Probab. 7 (1979), 75-84.

[13] Ordóñez Cabrera, M. and Sung, S.H., On complete convergence of weighted sums of random elements, Stochastic Anal. Appl. (To appear)

[14] Sung, S.H., Complete convergence for weighted sums of arrays of rowwise independent B-valued random variables, Stochastic Anal. Appl. 15 (1997), 255-267.

[15] Sung, S.H., Complete convergence for sums of arrays of random elements, Internat. J. Math. & Math. Sci. 23 (2000), 789-794.

[16] Wang, X.C., Rao, M.B. and Yang, X., Convergence rates on strong laws of large numbers for arrays of rowwise independent elements, Stochastic Anal. Appl. 11 (1993), 115-132.

Tien-Chung Hu, Department of Mathematics, Tsing Hua University, Hsinchu, Taiwan 30043, Republic of China Manuel Ordóñez Cabrera, Department of Mathematical Analysis, University of Sevilla, Sevila 41080, Spain Soo Hak Sun, Department of Applied Mathematics, Pai Chai University, Taejon 302-735, South Korea Andrei I. Volodin, Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, S4S 0A2 Canada; Research Institute of Mathematics and Mechanics, Kazan State University, Universitetskaya st. 17, Kazan, 420008, Russia

Received April 3, 2001