On the rate of complete convergence for weighted sums of arrays of random elements

ON THE RATE OF COMPLETE CONVERGENCE FOR WEIGHTED SUMS

OF ARRAYS OF RANDOM ELEMENTS

Soo Hak Sung and Andrei I. Volodin

Abstract. Let {V

, k ≥ 1, n ≥ 1} be an array of rowwise inde- pendent random elements which are stochastically dominated by a random variable X with E|X|

log

(|X|) < ∞ for some ρ >

0, α > 0, γ > 0, θ > 0 such that θ +α/γ < 2. Let {a

, k ≥ 1, n ≥ 1}

be an array of suitable constants. A complete convergence result is obtained for the weighted sums of the form P

k=1

a

V

.

1. Introduction

The concept of complete convergence of a sequence of random vari- ables was introduced by Hsu and Robbins [5] as follows. A sequence {U _n , n ≥ 1} of random variables converges completely to the constant θ

if X ∞

n=1

P (|U _n − θ| > ²) < ∞ for all ² > 0.

The classical Hsu-Robbins-Erd˝os theorem (Erd˝os [3, 4]) states that, for a sequence {X _n , n ≥ 1} of independent and identically distributed ran- dom variables, P _n

k=1 X _k /n converges completely to EX ₁ if and only if the variance of X ₁ is finite. Baum and Katz [2] obtained an elegant gen- eralization of Hsu-Robbins-Erd˝os theorem, namely they proved that, for r ≥ 1 and 1 ≤ t < 2r ≤ 2t, P _∞

n=1 n ^r−2 P (| P _n

k=1 (X _k − EX ₁ )| > n ^r/t ²) <

∞ for all ² > 0 if and only if E|X ₁ | ^t < ∞.

Many authors extended the above results to Banach space valued random elements, for example, see Ahmed et al. [1], Hu et al. [6, 7],

Received March 2, 2005. Revised June 10, 2005.

2000 Mathematics Subject Classification: Primary 60B12, 60F05, 60F15.

Key words and phrases: arrays of random elements, rowwise independence, weighted sums, complete convergence, rate of convergence, convergence in probability.

This work was supported by the Korea Research Foundation Grant (KRF-2003-

041-C00034).

Kuczmaszewska and Szynal [8], Sung [9], and Wang et al. [11]. A se- quence of Banach space valued random elements is said to converge completely to the 0 element in the Banach space if the corresponding sequence of norms converges completely to 0.

Hu, Rosalsky, Szynal and Volodin [7] presented a general result (cf.

Theorem 1 below) establishing complete convergence for the row sums of an array of rowwise independent but not necessarily identically dis- tributed Banach space valued random elements. Their result also spec- ified the corresponding rate of convergence. The Hu, Rosalsky, Szynal and Volodin [7] result unifies and extends previously obtained results in the literature in that many of them (for example, results of Hsu and Rob- bins [5], Hu et al. [6], Kuczmaszewska and Szynal [8], Sung [9], Volodin et al. [10], and Wang et al. [11]) follow from it.

In the following we assume that {V _nk , k ≥ 1, n ≥ 1} is an array of rowwise independent random elements in a real separable Banach space and {a _nk , k ≥ 1, n ≥ 1} is an array of constants. Denote

S _n ≡ X ∞ k=1

a _nk V _nk .

In the next theorem the weights a _nk are built into the array (that is, a _nk = 1 for all k and n).

Theorem 1. (Hu et al. [7]) Let {c _n , n ≥ 1} be a sequence of positive constants. Suppose that

X ∞ n=1

c _n X ∞ k=1

P (||V _nk || > ²) < ∞ for all ε > 0, (1)

X ∞ n=1

c _n à X ∞

k=1

E||V _nk || ^q

! _J

< ∞ for some 0 < q ≤ 2 and J ≥ 2, (2)

X ∞ k=1

V _nk → 0, ^P (3)

and

(4) if lim inf

n→∞ c _n = 0, then X ∞ k=1

P (||V _nk || > δ) = o(1) for some δ > 0.

Then

X ∞ n=1

c _n P (||S _n || > ²) < ∞ for all ε > 0.

It is implicitly assumed in Theorem 1 that the series S _n converges a.s.

The article Hu et al. [6] is devoted to presenting applications of The- orem 1 to obtain new complete convergence results. Theorem 2 gener- alizes results of Hsu and Robbins [5], Kuczmaszewska and Szynal [8], Sung [9], Wang et al. [11] in three directions, namely:

(i) Banach space valued random elements instead of random variables are considered.

(ii) An array rather than a sequence is considered.

(iii) The rate of convergence is obtained.

Theorem 2. (Hu et al. [6]). Suppose that the array {V _nk , k ≥ 1, n ≥ 1} is stochastically dominated by a random variable X. That is,

P (||V _nk || > x)

≤ CP (|X| > x) for all x > 0 and for all k ≥ 1 and n ≥ 1, where C is a positive constant. Assume that

sup

k≥1

|a _nk | = O(n ^−γ ) for some γ > 0, and X ∞

k=1

|a _nk | = O(n ^α ) for some α ∈ [0, γ).

If

E|X| 1+(1+α+β)/γ < ∞ for some β ∈ (−1, γ − α − 1], and S _n → 0, ^P

then X ∞

n=1

n ^β P (||S _n || > ²) < ∞ for all ε > 0.

The proof of Theorem 2 is rather complicated once it uses the Stieltjes integral techniques, summation by parts lemma and so on. The initial objective of an investigation resulted in the paper Ahmed et al. [1] was only to find a simpler proof. But it appears that they were able to estab- lish a more general result and with simpler proof. The result presented in Theorem 3 below is more general than the main result of Hu et al. [6], since rates of convergence for moving averages can be established, which cannot be proved using Theorem 2.

Theorem 3. (Ahmed et al. [1]) Suppose that the array {V _nk , k ≥

1, n ≥ 1} is stochastically dominated by a random variable X. Assume

that

sup

k≥1

|a _nk | = O(n ^−γ ) for some γ > 0, and

X ∞ k=1

|a _nk | = O(n ^α ) for some α < γ.

Let β be such that α + β 6= −1 and fix δ > 0 such that ^α _γ + 1 < δ ≤ 2. If E|X| ^ν < ∞, where ν = max{1 + 1 + α + β

γ , δ}, and

S _n → 0, ^P

then X ∞

n=1

n ^β P (||S _n || > ²) < ∞ for all ε > 0.

Theorem 3 was slightly generalized in Volodin et al. [10] as follows.

Theorem 4. (Volodin et al. [10]) Suppose that the array {V _nk , k ≥ 1, n ≥ 1} is stochastically dominated by a random variable X. Assume that

sup

k≥1

|a _nk | = O(n ^−γ ) for some γ > 0, and X ∞ k=1

|a _nk | ^θ = O(n ^α )

for some 0 < θ ≤ 2 and any α such that θ + ^α _γ < 2. Let β be such that α + β 6= −1 and fix δ > θ such that ^α _γ + θ < δ ≤ 2. If

E|X| ^ν < ∞, where ν = max{θ + 1 + α + β

γ , δ}, and S _n → 0, ^P

then X ∞

n=1

n ^β P (||S _n || > ²) < ∞ for all ε > 0.

If β < −1, then the conclusions of Theorems 3 and 4 are immedi-

ate and hence Theorems 3 and 4 are of interest only for β ≥ −1. In

particular, the case β = −1 is of special interest. Ahmed et al. [1] con-

jectured that when β = −1, the assumption E|X| ^ν < ∞ can be replaced

by E|X|

⁺¹ log ^ρ (|X|) < ∞ (ρ > 0) in Theorem 3. In the context of

Theorem 4 this conjecture should be rewritten as: when β = −1, the

assumption E|X| ^ν < ∞ can be replaced by the strictly weaker assump- tion E|X|

^+θ log ^ρ (|X|) < ∞ (ρ > 0). In this paper we give the positive answer on this conjecture.

It proves convenient to define log(x) = max{1, ln(x)}, where ln(x) de- notes the natural logarithm. The symbol C denotes a positive constant which is not necessarily the same one in each appearance, the symbol [x] denotes the greatest integer in x, and for a finite set A the symbol

#A denotes the number of elements in the set A.

2. Preliminaries

In this section, we present three lemmas which will be used to prove our main result.

Lemma 1. Let {a _nk , k ≥ 1, n ≥ 1} be an array of constants such that for some θ > 0, some α, and any n ≥ 1

X ∞ k=1

|a _nk | ^θ ≤ n ^α .

Let {φ(j), j ≥ 1} be an increasing sequence of positive numbers and I _nj =

½

k| 1

n ^γ φ(j + 1) < |a _nk | ≤ 1 n ^γ φ(j)

¾

, j ≥ 1, n ≥ 1, where γ is a constant. Then for any m ≥ 1

X m j=1

]I _nj ≤ n ^α+γθ φ ^θ (m + 1).

Proof. Really, X m j=1

]I _nj = X m j=1

X

k∈I

|a _nk | ^θ 1

|a _nk | ^θ

≤ n ^γθ X m j=1

φ ^θ (j + 1) X

k∈I

|a _nk | ^θ ≤ n ^α+γθ φ ^θ (m + 1).

Lemma 2. Let {V _nk , k ≥ 1, n ≥ 1} be an array of random ele-

ments which are stochastically dominated by a random variable X. Let

{a _nk , k ≥ 1, n ≥ 1} be an array of constants such that sup

k≥1

|a _nk | ≤ n ^−γ for some γ > 0 and

X ∞ k=1

|a _nk | ^θ ≤ n ^α for some θ > 0 and some α.

Then for any ² > 0 and all n ≥ 1 X ∞

k=1

P (ka _nk V _nk k > ²) ≤ Cn ^α X ∞ k=n

k ^γθ P (k <

¯ ¯

¯ ¯ X

²

¯ ¯

¯ ¯

1/γ

≤ k + 1).

Proof. In Lemma 1 consider φ(j) = j ^γ , j ≥ 1. Then I _nj =

½

k| 1

(n(j + 1)) ^γ < |a _nk | ≤ 1 (nj) ^γ

¾

for j ≥ 1 and n ≥ 1 and

X m j=1

]I _nj ≤ n ^α+γθ (m + 1) ^γθ .

Mention that the condition sup _k≥1 |a _nk | ≤ n ^−γ ensures us that, for any n ≥ 1, ∪ _j≥1 I _nj = {k|a _nk 6= 0}. It follows that

X ∞ k=1

P (ka _nk V _nk k > ²) = X ∞ j=1

X

k∈I

P (ka _nk V _nk k > ²)

≤ X ∞ j=1

X

k∈I

P (kV _nk k > ²(nj) ^γ )

≤ C X ∞ j=1

]I _nj P (| X

² | > (nj) ^γ )

= C X ∞ j=1

]I _nj X ∞ k=nj

P (k < | X

² | ^1/γ ≤ k + 1)

= C X ∞ k=n

P (k < | X

² | ^1/γ ≤ k + 1)

[

]

X

j=1

]I _nj

≤ C X ∞ k=n

P (k < | X

² | ^1/γ ≤ k + 1)n ^α+γθ ([ k

n ] + 1) ^γθ

≤ C2 ^γθ n ^α X ∞ k=n

P (k < | X

² | ^1/γ ≤ k + 1)k ^γθ .

Lemma 3. Let all the conditions of Lemma 2 be satisfied and α ≥ 0.

Then for all ² > 0 X ∞ k=1

P (ka _nk V _nk k > ²) ≤ CE

¯ ¯

¯ ¯ X

²

¯ ¯

¯ ¯

α γ

+θ

I(|X| > ²n ^γ ).

Proof. By Lemma 2 X ∞

k=1

P (ka _nk V _nk k > ²)

≤ Cn ^α X ∞ k=n

P µ

k <

¯ ¯

¯ ¯ X

²

¯ ¯

¯ ¯

1/γ

≤ k + 1

¶ k ^γθ

≤ C X ∞ k=n

P µ

k <

¯ ¯

¯ ¯ X

²

¯ ¯

¯ ¯

1/γ

≤ k + 1

¶

k ^γθ+α (since α ≥ 0)

≤ CE| X

² |

^+θ I(|X| > ²n ^γ ).

3. Main result

In this section, we state and prove our main result.

Theorem 5. Let {V _nk , k ≥ 1, n ≥ 1} be an array of rowwise indepen- dent random elements which are stochastically dominated by a random variable X. Let {a _nk , k ≥ 1, n ≥ 1} be an array of constants such that

sup

k≥1 |a _nk | = O(n ^−γ ) for some γ > 0 and

X ∞ k=1

|a _nk | ^θ = O(n ^α ) for some α > 0 and θ > 0 such that θ + α

γ < 2.

Assume that

S _n ≡ X ∞ k=1

a _nk V _nk → 0. ^P

If E|X|

^+θ log ^ρ (|X|) < ∞ for some ρ > 0, then X ∞

n=1

1

n P (kS _n k > ²) < ∞ for all ² > 0.

Proof. Without loss of generality, we may assume that sup

k≥1

|a _nk | ≤ n ^−γ and P ^∞

k=1

|a _nk | ^θ ≤ n ^α . For any n ≥ 1 let

V _nk ⁽¹⁾ = V _nk I(ka _nk V _nk k ≤ 1) and

V _nk ⁽²⁾ = V _nk I(ka _nk V _nk k > 1), 1 ≤ k < ∞.

Then

S _n = X ∞ k=1

a _nk V _nk = X 2 p=1

X ∞ k=1

a _nk V _nk ^(p) = X 2 p=1

S _n ^(p) , say.

To prove the theorem, it suffices to show that for p = 1 and 2:

X ∞ n=1

1

n P (kS _n ^(p) k > ²) < ∞ for all ² > 0.

To do this, we apply Theorem 1 with c _n = 1/n to the random ele- ments a _nk V _nk ^(p) , p = 1, 2.

Then

p=1,2 max X ∞ n=1

1 n

X ∞ k=1

P (ka _nk V _nk ^(p) k > ²)

≤ X ∞ n=1

1 n

X ∞ k=1

P (ka _nk V _nk k > ²)

≤ C X ∞ n=1

n ^α−1 X ∞ k=n

k ^γθ P (k < | X

² | ^1/γ ≤ k + 1) (by Lemma 2)

= C X ∞ k=1

k ^γθ P (k < | X

² | ^1/γ ≤ k + 1) X k n=1

n ^α−1

≤ C X ∞ k=1

k ^α+γθ P (k < | X

² | ^1/γ ≤ k + 1) (since α > 0)

≤ CE| X

² |

^+θ < ∞.

Hence (1) is satisfied for both series.

By Lemma 3, we have for any ² > 0 P (k

X ∞ k=1

a _nk V _nk I(ka _nk V _nk k > 1)k > ²) ≤ P (∪ ^∞ _k=1 ka _nk V _nk k > 1)

≤ X ∞ k=1

P (ka _nk V _nk k > 1)

≤ CE|X|

^+θ I(|X| > n ^γ ) = o(1), since E|X|

^+θ < ∞. Hence S _n ⁽²⁾ → 0. By the hypothesis S ^P _n → 0 and ^P S _n ⁽¹⁾ = S _n − S _n ⁽²⁾ , we have S _n ⁽¹⁾ → 0. We conclude that condition (3) from ^P Theorem 1 is satisfied for both series.

The condition (4) from Theorem 1 with δ = 1 is obviously satisfied for the first series, since

X ∞ k=1

P (ka _nk V _nk ⁽¹⁾ k > 1) = 0.

For the second series we have by Lemma 3 that X ∞

k=1

P (ka _nk V _nk ⁽²⁾ k > 1) = X ∞ k=1

P (ka _nk V _nk k > 1)

≤ CE|X|

^+θ I(|X| > n ^γ ) = o(1).

Hence, condition (4) is satisfied for both series.

Finally, we check condition (2) from Theorem 1 for both series. To

do this, we introduce the following notations. For t > 0 such that

0 < tθ < ρ and any n ≥ 1, let A _n =

½

k| |a _nk | ≤ 1 n ^γ log ^t (n)

¾

, B _n =

½

k| 1

n ^γ log ^t (n) < |a _nk | ≤ 1 n ^γ

¾ . Next, for any n ≥ 1 let

a ⁽¹⁾ _nk =

½ a _nk if k ∈ A _n

0 otherwise , a ⁽²⁾ _nk =

½ a _nk if k ∈ B _n 0 otherwise.

Let r = ^α _γ + θ < 2. First we mention that

p=1,2 max X ∞ k=1

Eka ⁽¹⁾ _nk V _nk ^(p) k ^r ≤ X

k∈A

Eka _nk V _nk k ^r

≤ C

µ 1

n ^γ log ^t (n)

¶ _α/γ

E|X| ^α/γ+θ X ∞ k=1

|a _nk | ^θ

≤ C log ^−tα/γ (n)E|X| ^α/γ+θ . Hence we have that

(5) X ∞ k=1

Eka ⁽¹⁾ _nk V _nk ⁽¹⁾ k ² ≤ X ∞ k=1

Eka ⁽¹⁾ _nk V _nk ⁽¹⁾ k ^r ≤ C log ^−tα/γ (n)E|X| ^α/γ+θ

and (6)

X ∞ k=1

Eka ⁽¹⁾ _nk V _nk ⁽²⁾ k ^r ≤ C log ^−tα/γ (n)E|X| ^α/γ+θ .

In order to verify condition (2) for other cases, put B _nj =

½

k| 1

n ^γ log ^t (j + 1) < |a _nk | ≤ 1 n ^γ log ^t (j)

¾ .

Then {B _nj , 1 ≤ j ≤ n − 1} are disjoint, ∪ ⁿ⁻¹ _j=1 B _nj = B _n , and by Lemma 1 with φ(j) = log ^t (j), ]B _n ≤ n ^α+γθ log ^tθ (n). We can estimate

X ∞ k=1

Eka ⁽²⁾ _nk V _nk ⁽²⁾ k ^r = X

k∈B

|a _nk | ^r EkV _nk k ^r I µ

kV _nk k > 1

|a _nk |

¶

≤ C X

k∈B

|a _nk | ^r E|X| ^r I(|X| > n ^γ )

≤ CE|X| ^r I(|X| > n ^γ )(n ^−γ ) ^r ]B _n

≤ CE|X| ^r I(|X| > n ^γ ) log ^tθ (n) (7)

≤ CE|X| ^r log ^ρ (|X|)I(|X| > n ^γ ) log ^tθ−ρ (n)

≤ CE|X| ^r log ^ρ (|X|) log ^tθ−ρ (n).

Next, we estimate the remaining part in the following way X ∞

k=1

Eka ⁽²⁾ _nk V _nk ⁽¹⁾ k ²

=

n−1 X

j=1

X

k∈B

Eka _nk V _nk k ² I(ka _nk V _nk k ≤ 1)

≤

n−1 X

j=1

X

k∈B

1

(n ^γ log ^t (j)) ² EkV _nk k ² I(kV _nk k ≤ n ^γ log ^t (j + 1))

≤ C

n−1 X

j=1

]B _nj n ^−2γ log ^−2t (j)EX ² I(|X| ≤ n ^γ log ^t (j + 1))

+ C

n−1 X

j=1

]B _nj P (|X| > n ^γ log ^t (j + 1))

= I ₁ + I ₂ , say.

Here we used the fact that if a random variable Y is stochastically dom- inated by a random variable X, then for all s > 0 and b > 0

E|Y | ^s I(|Y | ≤ b) ≤ CE|X| ^s I(|X| ≤ b) + Cb ^s P (|X| > b).

Let µ = 2 − ^α _γ − θ > 0. Since _log ^x

(x) ↑ as x → ∞, x ^µ

log ^ρ (x) ≤ (n ^γ log ^t (j + 1)) ^µ

log ^ρ (n ^γ log ^t (j + 1)) ≤ C n ^γµ log ^tµ (j) log ^ρ (n) , if x ≤ n ^γ log ^t (j + 1). Then

I ₁ ≤ C

n−1 X

j=1

]B _nj n ^−2γ log ^−2t (j)E|X| ^r log ^ρ (|X|) n ^γµ log ^tµ (j) log ^ρ (n)

= C n ^γµ−2γ

log ^ρ (n) E|X| ^r log ^ρ (|X|)

n−1 X

j=1

log ^tµ−2t (j)]B _nj

≤ C n ^γµ−2γ

log ^ρ (n) E|X| ^r log ^ρ (|X|)

n−1 X

j=1

]B _nj (since tµ − 2t < 0)

≤ CE|X| ^r log ^ρ (|X|) log ^tθ−ρ (n).

Clearly, I ₂ is dominated by

CP (|X| > n ^γ )

n−1 X

j=1

]B _nj ≤ CP (|X| > n ^γ )n ^α+γθ log ^tθ (n)

≤ CE|X| ^r log ^ρ (|X|) log ^tθ−ρ (n).

Hence (8)

X ∞ k=1

Eka ⁽²⁾ _nk V _nk ⁽¹⁾ k ² ≤ CE|X| ^α/γ+θ log ^ρ (|X|) log ^tθ−ρ (n).

Take J such that J(ρ − tθ) ≥ 2 and Jtα/γ ≥ 2. We have by (5) and (8) that

X

1 n

µ X

Eka

V

k

¶

= X

1 n

µ X

Eka

V

k

+ X

k=1

Eka

V

k

¶

≤ X

1 n

µ

CE|X|

log

(n) + CE|X|

log

(|X|) log

(n)

¶

≤ C X

1 n log

(n) ,

since E|X| ^α/γ+θ log ^ρ (|X|) < ∞. Hence for the first series condition (2) from Theorem 1 is satisfied with q = 2.

Next, by (6) and (7) we have that X ∞

n=1

1 n

µ X _∞

k=1

Eka _nk V _nk ⁽²⁾ k ^r

¶ _J

= X ∞ n=1

1 n

µ X _∞

k=1

Eka ⁽¹⁾ _nk V _nk ⁽²⁾ k ^r + X ∞ k=1

Eka ⁽²⁾ _nk V _nk ⁽²⁾ k ^r

¶ _J

≤ X ∞ n=1

1 n

µ

CE|X| ^α/γ+θ log ^−tα/γ (n)+CE|X| ^α/γ+θ log ^ρ (|X|) log ^tθ−ρ (n)

¶ _J

≤ C X ∞ n=1

1 n log ² (n) .

Hence for the second series condition (2) from Theorem 1 is satisfied with q = r.

Therefore all conditions from Theorem 1 are satisfied for both series, and so the proof is complete.

Acknowledgements. We would like to thank the referee for his careful reading of this paper and for his helpful comments.

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Soo Hak Sung

Department of Applied Mathematics Pai Chai University

Taejon 302-735, Korea E-mail: [email protected] Andrei I. Volodin

Department of Mathematics and Statistics University of Regina

Regina, Saskatchewan, S4S 0A2, Canada

E-mail: [email protected]