On the complete convergence for sequences of coordinatewise negatively associated random vectors in Hilbert spaces

We develop the BaumKatz theorem for sequences of coordinatewise negatively associated random vectors in real separable Hilbert spaces. We also show that the concept of coordinatewise negative association is more general than the concept of negative association of Ko et al. (2009) 9. Moreover, some related results still hold for this concept. Illustrative examples are provided.

On the complete convergence for sequences of

coordinatewise negatively associated random vectors in

Hilbert spaces

Nguyen Van Huan∗

Department of Mathematics and Applications, Saigon University, Ho Chi Minh City,

Vietnam

Nguyen Van Quang

Department of Mathematics, Vinh University, Nghe An Province, Vietnam

Nguyen Tran Thuan

Department of Mathematics, Vinh University, Nghe An Province, Vietnam

Abstract

We develop the Baum-Katz theorem for sequences of coordinatewise negatively associated random vectors in real separable Hilbert spaces We also show that the concept of coordinatewise negative association is more general than the concept of negative association of Ko et al (2009) [9] Moreover, some related results still hold for this concept Illustrative examples are provided

Keywords: Coordinatewise negatively associated, Real separable Hilbert space, Coordinatewise weakly bounded

2000 MSC: 60F15, 60B11, 60B12

∗ Corresponding author: Department of Mathematics and Applications, Saigon Uni-versity, 273 An Duong Vuong Street, Ward 3, District 5, Ho Chi Minh City, Vietnam; Tel.: +84 917918008 (Mobile), +84 839381913 (Office); fax: +84 838305568.

Email addresses: vanhuandhdt@yahoo.com (Nguyen Van Huan),

nvquang@hotmail.com (Nguyen Van Quang), tranthuandhv@gmail.com (Nguyen Tran Thuan)

1 Introduction

Hsu and Robbins [7] introduced the concept of complete convergence and proved that the sequence of arithmetic means of independent, identically distributed (i.i.d.) random variables converges completely to the expected value of the variables, provided their variance is finite The necessity was proved by Erd¨os [4, 5] The result of Hsu-Robbins-Erd¨os is a fundamental theorem in probability theory and was later generalized and extended during

a process which led to the now classical paper by Baum and Katz [3] Theorem 1.1 ([3]) Let r, α be real numbers (r > 1; α > 1/2; αr > 1), and let {Xn, n > 1} be a sequence of i.i.d random variables with zero means Then the following three statements are equivalent:

(a) E|X1|r < ∞

(b)

∞

X

n=1

nαr−2 P

n

X

k=1

Xk

> εnα< ∞ for every ε > 0

(c)

∞

X

n=1

nαr−2 Psup

k>n

1

kα

k

X

l=1

Xl

> ε< ∞ for every ε > 0

This result has been extensively studied for many classes of dependent random variables For negatively associated random variables, we refer

to Shao [16], Kuczmaszewska [10], Baek et al [2], Kuczmaszewska and Lagodowski [11], and other authors

In this paper, we discuss the concept of negative association for random vectors in a real separable Hilbert space and develop the Baum-Katz theorem for sequences of coordinatewise negatively associated random vectors which

by our knowledge have not yet been studied in the literature

The concept of negative association for random variables was introduced

by Alam and Saxena [1] and carefully studied by Joag-Dev and Proschan [8]

A finite family {Yi, 1 6 i 6 n} of random variables is said to be negatively associated (NA) if for any disjoint subsets A, B of {1, 2, , n} and any real coordinatewise nondecreasing functions f on R|A|, g on R|B|,

Cov f (Yi, i ∈ A), g(Yj, j ∈ B)
6 0 whenever the covariance exists, where |A| denotes the cardinality of A An infinite family of random variables is NA if every finite subfamily is NA

As in Ko et al [9], a finite family {Xi, 1 6 i 6 n} of Rd-valued random vectors is said to be NA if for any disjoint subsets A, B of {1, 2, , n} and any real coordinatewise nondecreasing functions f on R|A|d, g on R|B|d,

Cov f (Xi, i ∈ A), g(Xj, j ∈ B)
6 0 whenever the covariance exists An infinite family of Rd-valued random vectors is NA if every finite subfamily is NA

Let H be a real separable Hilbert space with the norm k · k generated

by an inner product h·, ·i, let {ej, j > 1} be an orthonormal basis in H, let X be an H-valued random vector, and hX, eji will be denoted by X(j)

Ko et al [9] introduced the concept of H-valued NA sequence as follows Definition 1.2 ([9]) A sequence {Xn, n > 1} of H-valued random vectors

is said to be NA if for any d > 1, the sequence Xn(1), Xn(2), , Xn(d)
, n > 1

of Rd-valued random vectors is NA

In the following definition, we present another concept of negative association for H-valued random vectors which is more general than the concept of Ko et al [9]

Definition 1.3 A sequence {Xn, n > 1} of H-valued random vectors is said to be coordinatewise negatively associated (CNA) if for each j > 1, the sequence {Xn(j), n > 1} of random variables is NA

Obviously, if a sequence of H-valued random vectors is NA, then it is CNA However, the reverse is not true in general In the following example,

we derive an Rd-valued CNA sequence which is not NA

Example 1.4 Let d be an integer (d > 2), and let {Yn, n = 1, 2, , d} be

a sequence of random variables which is not NA We consider a sequence

Xn = (Xn(1), Xn(2), , Xn(d)), n = 1, 2, , d of Rd-valued random vectors as follows: For each n = 1, 2, , d, Xn(n) = Yn and for each j = 1, 2, , d, {Xn(j), n = 1, 2, , d} is a sequence of independent random variables Then the sequence {Xn, n = 1, 2, , d} is CNA, but it is not NA

Ko et al [9] obtained the almost sure convergence for sequences of H-valued NA random vectors The key tool for proving their result is the maximal inequality provided by the following lemma

Lemma 1.5 ([9], Lemma 3.3) Let {Xn, n > 1} be a sequence of H-valued

NA random vectors with EXn = 0 and EkXnk2

< ∞, n > 1 Then, we have E

 max

16k6n

k

X

i=1

Xi 26

n

X

i=1

EkXik2, n > 1 (1.1)

Let us note that there is a misprint in Lemma 3.3 of Ko et al [9], as the following example shows

Example 1.6 Let {X, Xn, n > 1} be a sequence of i.i.d random variables with zero means and finite second moments Then

E



max

16k62

k

X

i=1

Xi

2

= E max{|X1|, |X1+ X2|}
2

= E

|X1| + |X1+ X2| +|X1| − |X1+ X2|

2

2

= EX12+1

2EX

2

2 +1

2E|X

2

2 + 2X1X2|

> EX12+ 1

2EX

2

2 + 1

2|EX2

2 + 2E(X1X2)|

= EX12+ EX22

It is not hard to check that if X is a symmetric random variable taking values

in {−1; 1}, then E|X2

2 + 2X1X2| 6= |EX2

2 + 2E(X1X2)| So (1.1) fails

The following lemma is an improvement of Lemma 1.5 and was proved

by Shao [16] in the case of NA random variables

Lemma 1.7 Let {Xn, n > 1} be a sequence of H-valued CNA random vectors with EXn= 0 and EkXnk2 < ∞, n > 1 Then, we have

E

 max

16k6n

k

X

i=1

Xi 26 2

n

X

i=1

EkXik2, n > 1

Remark 1.8 In view of Lemma 1.7 and the proof of Theorem 3.4 of Ko et

al [9], it is interesting to observe that Theorem 3.4 in [9] (see also Miao [14, Theorems 3.2 and 3.3], Thanh [17, Theorems 2.2 and 3.1]) does not only hold for H-valued NA sequences, but also holds for H-valued CNA sequences

Let {X, Xn, n > 1} be a sequence of H-valued random vectors We consider the following inequality

C1P(|X(j)| > t) 6 1

n

n

X

k=1

P(|Xk(j)| > t) 6 C2P(|X(j)| > t) (1.2)

If there exists a positive constant C1 (C2) such that the left-hand side (right-hand side) of (1.2) is satisfied for all j > 1, n > 1 and t > 0, then the sequence {Xn, n > 1} is said to be coordinatewise weakly lower (upper) bounded by X The sequence {Xn, n > 1} is said to be coordinatewise weakly bounded by X if it is both coordinatewise weakly lower and upper bounded

by X Note that (1.2) is, of course, automatic with X = X1 and C1 = C2 = 1

if {Xn, n > 1} is a sequence of identically distributed random vectors

In the rest of the paper, the symbol C will denote a generic positive constant which is not necessarily the same one in each appearance

2 Main results and discussions

Theorem 2.1 Let r, α be positive real numbers (1 6 r < 2; αr > 1), and let {Xn, n > 1} be a sequence of H-valued CNA random vectors with zero means Suppose that {Xn, n > 1} is coordinatewise weakly upper bounded by

a random vector X If

∞

X

j=1

then

∞

X

n=1

nαr−2 P max

16k6n

k

X

l=1

Xl > εnα< ∞ for every ε > 0 (2.2) Remark 2.2 From (2.2) and Lemma 4 of Lai [12] we have

∞

X

n=1

nαr−2 Psup

k>n

1

kα

k

X

l=1

Xl > ε< ∞ for every ε > 0

Then by the Kronecker lemma,

P



sup

k>n

1

kα

k

X

l=1

Xl > ε= o n1−αr
for every ε > 0

Therefore, the conclusion (2.2) describes the rate of convergence in the strong law of large numbers

Remark 2.3 Theorem 2.1 still holds if the condition that {Xn, n > 1}

is coordinatewise weakly upper bounded by X is replaced by the following weaker condition:

1

n

n

X

k=1

∞

X

j=1

P(|Xk(j)| > t) 6 C

∞

X

j=1

P(|X(j)| > t), n > 1, t > 0

Remark 2.4 In the case 0 < r < 1, the implication (2.1) ⇒ (2.2)
of Theorem 2.1 holds without coordinatewise negative association and mean zero conditions on the random vectors

Under the assumptions of Theorem 2.1, (2.1) implies (2.2) A natural question is whether or not the converse is true A negative answer to this question will be given in following example

Example 2.5 We consider the space `2 consisting of square summable real sequences x = {xk, k > 1} with norm kxk = P∞

k=1x2 k

1/2

Let {Xn, n > 1} be a sequence of `2-valued i.i.d random vectors such that P Xn(j) =

±j−1/r
= 1/2 for all n > 1 and j > 1 It is well known that the space `2 is

of type 2, and so it is of type p for all r < p 6 2 (for details see Pisier [15]) Then, for every ε > 0,

∞

X

n=1

nαr−2 P max

16k6n

k

X

l=1

Xl > εnα

6 C

∞

X

n=1

nα(r−p)−2E max

16k6n

k

X

l=1

Xl 

p

6 C

∞

X

n=1

nα(r−p)−2

n

X

k=1

EkXkkp

= C

∞

X

n=1

nα(r−p)−1 EkX1kp = C

∞

X

n=1

nα(r−p)−1 E

∞

X

j=1

1

j2/r

p/2

< ∞,

and therefore (2.2) holds However, (2.1) fails since

∞

X

j=1

E|X1(j)|r =

∞

X

j=1

1

j = ∞.

The following theorem provides sufficient conditions for (2.1) to hold Theorem 2.6 Let r, α be positive real numbers such that αr > 1, and let {Xn, n > 1} be a sequence of H-valued CNA random vectors with zero means Suppose that {Xn, n > 1} is coordinatewise weakly bounded by a random vector X with

∞

X

j=1

E |X(j)|rI(|X(j)| 6 1)
< ∞ (2.3)

If

∞

X

j=1

∞

X

n=1

nαr−2 P max

16k6n

k

X

l=1

Xl(j)

> εnα< ∞ for every ε > 0, (2.4) then (2.1) holds

Obviously, if the condition (2.3) is not satisfied, then the conclusion (2.1) fails The following example shows that, in Theorem 2.6, we cannot remove (2.3) or even replace it by the weaker condition E |X(j)|rI(|X(j)| 6 1)
= o(1) as j → ∞

Example 2.7 Let p, r be positive real numbers such that r < p 6 2 We consider the sequence {Xn, n > 1} in Example 2.5 Then, for every ε > 0,

we have

∞

X

j=1

∞

X

n=1

nαr−2 P max

16k6n

k

X

l=1

Xl(j)

> εnα

6 C

∞

X

j=1

∞

X

n=1

nα(r−p)−2 E max

16k6n

k

X

l=1

Xl(j)

p

6 C

∞

X

j=1

∞

X

n=1

nα(r−p)−2

n

X

k=1

E|Xk(j)|p

(since R is of type p)

= C

∞

X

j=1

∞

X

n=1

nα(r−p)−1 E|X1(j)|p = C

∞

X

n=1

nα(r−p)−1

∞

X

j=1

1

jp/r < ∞,

so that (2.4) holds We also see that E |X(j)|rI(|X(j)| 6 1)
= 1/j = o(1) However, the conclusion (2.1) fails

Remark 2.8 Let r, s be positive real numbers such that s < r < 2, and let {Xn, n > 1} be a sequence of `2-valued i.i.d random vectors with

P Xn(j) = ±j−1/s
= 1/2, n > 1 Then by using the same arguments as

in Example2.7, we can show that the conditions (2.3) and (2.4) are satisfied Theorem 2.6 ensures that (2.1) holds

The example below shows that Theorem2.6 can fail if the series in (2.4) diverges

Example 2.9 Let r > 2, let {Yn, n > 1} be a sequence of i.i.d symmetric random variables such that E|Y1|r/2 = ∞ and |Yn| < ∞ for all n > 1 For each n > 1, set

Xn(j) = YnI(j − 1 6 |Yn|r/2< j), j > 1; Xn=

∞

X

j=1

Xn(j)ej,

where {ej, j > 1} is an orthonormal basis in `2 Then, for each j > 1, {Xn(j), n > 1} is a sequence of i.i.d symmetric random variables Moreover, since

∞

X

j=1

Xn(j)
2 = Yn2 < ∞, EXn = E

∞

X

j=1

Xn(j)ej= 0, n > 1,

{Xn, n > 1} is a sequence of `2-valued CNA random vectors with zero means and coordinatewise weakly bounded by X1 Now

∞

X

j=1

E |X1(j)|rI(|X1(j)| 6 1)
= E |X(1)

1 |rI(|X1(1)| 6 1)
+ E |X1(2)|rI(|X1(2)| 6 1)
< ∞ and so (2.3) is satisfied Let α = 2/r, we will show that the series in (2.4) diverges Without loss of generality, assume that ε = 1 Then, we have

∞

X

j=1

∞

X

n=1

nαr−2 P

 max

16k6n

k

X

l=1

Xl(j)

> εnα



>

∞

X

j=1

∞

X

n=1

P |X1(j)| > n2/r
=

∞

X

n=1

∞

X

j=1

P |X1(j)|r/2> n

>

∞

X

n=1

∞

X

j=n+2

P j − 1 6 |Y1|r/2 < j
> E|Y1|r/2− 2 = ∞

Thus, (2.3) does not imply (2.4) Moreover, in this case

∞

X

j=1

E|X1(j)|r

= E

∞

X

j=1

|X1(j)|r

= E|Y1|r = ∞

Hence, (2.1) fails

Note that if r 6 2, then the condition (2.1) is stronger than the condition

However, in the special case when H is finite dimensional, (2.1) and (2.5) are equivalent Moreover, we have the following corollary

Corollary 2.10 Let r, α be positive real numbers (1 6 r < 2; αr > 1), let H

be a finite dimensional real Hilbert space, and let {Xn, n > 1} be a sequence

of H-valued CNA random vectors with zero means Suppose that {Xn, n > 1}

is coordinatewise weakly bounded by a random vector X Then (2.1), (2.2), (2.4), (2.5) are equivalent

3 Proofs

Proof of Lemma 1.7 In view of the proof of Lemma 4 of Matu la [13],

we have

E



max

16k6n

k

X

i=1

Xi

2

= E

 max

16k6n

∞

X

j=1

D k

X

i=1

Xi, ejE

2

6 E

 ∞

X

j=1

max

16k6n

D k

X

i=1

Xi, ejE2



=

∞

X

j=1

E

 max

n

max

16k6n

k

X

i=1

Xi(j)

2

;

 max

16k6n

k

X

i=1

− Xi(j)
2o

6

∞

X

j=1

E

 max

16k6n

k

X

i=1

Xi(j)

2

+

∞

X

j=1

E

 max

16k6n

k

X

i=1

− Xi(j)
2

6 2

∞

X

j=1

n

X

i=1

E Xi(j)

2

= 2

n

X

i=1

EkXik2

The proof is completed

To prove Theorem2.1, we need the following lemma Note that the proof

of this lemma is quite simple if H is finite dimensional

Lemma 3.1 Let p, r, α be positive real numbers (r < p; αr > 1), and let X

be an H-valued random vector satisfying (2.1) Then

∞

X

j=1

∞

X

n=1

nα(r−p)−1 E (X(j))pI(|X(j)| 6 nα)
< ∞

Proof We have

∞

X

j=1

∞

X

n=1

nα(r−p)−1E (X(j))pI(|X(j)| 6 nα)

=

∞

X

j=1

∞

X

n=1

nα(r−p)−1 E (X(j))pI(|X(j)| 6 1)

+

∞

X

j=1

∞

X

n=1

nα(r−p)−1 E (X(j))pI(1 < |X(j)| 6 nα)
= I1+ I2

It follows from (2.1) that I1 6

∞

P

j=1

E|X(j)|r P∞

n=1

nα(r−p)−1< ∞ Now we prove

I2 < ∞ Indeed,

I2 = p

∞

X

j=1

∞

X

n=1

nα(r−p)−1

Z n α

0

xp−1P |X(j)| I(1 < |X(j)

| 6 nα) > x
dx

6 p

∞

X

j=1

∞

X

n=1

nα(r−p)−1

Z 1

0

xp−1P |X(j)| > 1
dx

+ p

∞

X

j=1

∞

X

n=1

nα(r−p)−1

Z n α

1

xp−1P |X(j)| > x
dx

6

∞

X

j=1

E|X(j)|r

∞

X

n=1

nα(r−p)−1+ C

∞

X

j=1

∞

X

n=1

nα(r−p)−1

n

X

k=1

P |X(j)| > kα
kpα−1

= C + C

∞

X

j=1

I3(j),

I3(j) =

∞

X

k=1

P |X(j)| > kα
kpα−1

∞

X

n=k

nα(r−p)−1

6 C

∞

X

k=1

P |X(j)| > kα
kpα−1

Z ∞

k

1

xα(p−r)+1dx

α(p − r)

∞

X

k=1

kαr−1P |X(j)| > kα

= C

∞

X

k=1

kαr−1

∞

X

n=k

P nα < |X(j)| 6 (n + 1)α

6 C

∞

X

n=1

nαrP nαr < |X(j)|r

6 (n + 1)αr
6 C E|X(j)|r Since the last constant C depends only on p, r and α, we obtain I2 < ∞

Proof of Theorem 2.1 For n, k, j > 1, set

Ynk(j) = Xk(j)I(|Xk(j)| 6 nα) + nαI(Xk(j) > nα) − nαI(Xk(j)< −nα);

Ynk =

∞

X

j=1

Ynk(j)ej

Then for every ε > 0,

∞

X

n=1

nαr−2 P

 max

16k6n

k

X

l=1

Xl > εnα



=

∞

X

n=1

nαr−2 P max

16k6n

k

X

l=1

∞

X

j=1

Xl(j)ej > εnα

6

∞

X

n=1

nαr−2 P max

16k6nmax

j>1 |Xk(j)| > nα

+

∞

X

n=1

nαr−2 P max

16k6n

k

X

l=1

∞

X

j=1

Ynl(j)ej > εnα

In this paper, we discuss the concept of negative association for random vectors in a real separable Hilbert space and develop the Baum-Katz theorem for sequences of coordinatewise. .. 2.4 In the case < r < 1, the implication (2.1) ⇒ (2.2)
of Theorem 2.1 holds without coordinatewise negative association and mean zero conditions on the random vectors

Under the. .. Xn(d)
, n >

of Rd-valued random vectors is NA

In the following definition, we present another concept of negative association for H-valued random vectors which is more