Weak laws of large numbers for weighted coordinatewise pairwise NQD random vectors in Hilbert spaces

https://doi.org/10.4134/JKMS.j180217 pISSN: 0304-9914 / eISSN: 2234-3008

WEAK LAWS OF LARGE NUMBERS FOR WEIGHTED COORDINATEWISE PAIRWISE NQD RANDOM VECTORS

IN HILBERT SPACES

Dung Van Le, Son Cong Ta, and Cuong Manh Tran

Abstract. In this paper, we investigate weak laws of large numbers for weighted coordinatewise pairwise negative quadrant dependence random vectors in Hilbert spaces in the case that the decay order of tail probability is r for some 0 < r < 2. Moreover, we extend results concerning Pareto- Zipf distributions and St. Petersburg game.

1. Introduction

Let {X _n ; n ≥ 1} be the player’s gains in a sequence of independent repeti- tions of the St. Petersburg game, i.e., a sequence of iid random variables with the common distribution

P (X = 2 ^k ) = 2 ^−k for k = 1, 2, . . . .

For the total gain, S n := X 1 + · · · + X n in n games, Feller [4] proved that S _n /(n log n) → 1 in probability as n → ∞.

A. Gut [6] gave weak laws of large numbers for a generalized St. Petersburg game. Note that these games are formulated by nonnegative random variables with infinite means. On the other hand, Adler [1] got rid of the identically distributed condition with respect to independent Pareto-Zipf distributions. He studied weighted laws of large numbers for each model, respectively. Recently, Nakata [16] obtained some weak laws of large numbers for weighted sums of independent random variables in the case that the decay order of tail probability is r (0 < r ≤ 1). Dung et al. [3] gave weak laws of large numbers for sequences of independent random variables with infinite rth moments (0 < r < 2).

Received April 3, 2018; Revised August 10, 2018; Accepted August 24, 2018.

2010 Mathematics Subject Classification. Primary 60F15, 60G50.

Key words and phrases. infinite moments, weak law of large numbers, coordinatewise pairwise NQD random vectors, Hilbert spaces, weighted sums.

This research has been supported by the Vietnam National University, Hanoi (grant no.

QG.16.09).

c

2019 Korean Mathematical Society

457

Lehmann [12] introduced the notion of negative quadrant dependence (NQD):

Two random variables X 1 and X 2 is called NQD if

P (X 1 > x 1 , X 2 > x 2 ) ≤ P (X 1 > x 1 )P (X 2 > x 2 )

for all real numbers x ₁ , x ₂ . A sequence of random variables {X _n , n ≥ 1} is said to be pairwise NQD if every pair of random variables in the sequence is NQD. It is easy to see that a pairwise NQD sequence of random variables is much weaker than the NA one [9]. In many statistics and mechanic models, a pairwise NQD assumption among the random variables in the models is more reasonable than an independence assumption, so many statisticians have investigated this topic with interest (see [13–15, 17]).

Let H be a real separable Hilbert space with the norm k · k generated by an inner product h·, ·i and let {e j , j ∈ B} be an orthonormal basis in H. In this paper, we investigate weak laws of large numbers for the weighted sum

(1) S _n =

m

n

X

j=1

a _nj X _j ,

where (a nj ; 1 ≤ j ≤ m n , n ≥ 1) is an array of real numbers, {X n ; n ≥ 1} is a sequence of coordinatewise pairwise NQD random vectors in Hilbert spaces which satisfies that

(2) X

j∈B

P (|hX _n , e _j i| > x)  x ^−r for a fixed 0 < r < 2.

It is easy to see that if the cardinality |B| of B is finite, random vectors X n

that fulfill (2), then EkX n k ^r = ∞. Moreover, (2) is equivalent to P (kX n k > x)  x ^−r for a fixed 0 < r < 2.

Many random evolutions and also statistical procedures such as parametric or nonparametric estimation of regression with fixed design, produce statistics of type (1). One example is the nonlinear regression model

y(x) = f (x) + ξ(x),

where f (x) is an unknown function and ξ(x) is the noise. Now, we fix the design points x n1 , . . . , x nn and we get

y ni = f (x ni ) + ξ i ,

where ξ i is a centered sequence of random variables. The nonparametric es- timator of f (x) is defined to be b f _n (x) = P n

i=1 w _ni (x)y _ni , where the weight functions w ni (x) = w ni (x, x n ) depend both on x and the design points x n = {x n1 , . . . , x nn }. It is obvious that b f n (x) − E b f n (x) is of the type (1).

We shall also see that the asymptotic behavior of the sum of variables of the form

X _k =

∞

X

i=−∞

a _k+i ξ _i

can be obtained by studying the sum of the type (1).

Methods for Hilbert space valued random vectors might also help to analyze nonlinear statistics of real valued data.

For example, we will consider general bivariate and degenerate von Mises- statistics (V-statistics). Let h : R ² → R be a symmetric, measurable function.

We call

V n := 1 b ² _n

n

X

i,j=1

h(X i , X j )

be V-statistic with kernel h. The kernel and related V-statistic are called degenerate, if E(h(x, X _i )) = 0 for all x ∈ R. Furthermore, we assume that h is Lipschitz-continuous and positive definite. By Sun’s version of Mercers theorem, we have under these conditions a representation

h(x, y) =

∞

X

l=1

λφ _l (x)φ _l (y)

for orthonormal eigenfuctions (φ _l ) _l∈N with the following properties: Eφ _l (X _n ) = 0, Eφ ² _l (X _n ) = 1 for all l ∈ N, and λ l ≥ 0 for all l ∈ N, P ∞

l=1 λ _l < ∞ (see [2]).

Let H be Hibert space of real-valued sequences y = (y _l ) _l∈N equipped with the inner product

hy, zi :=

∞

X

l=1

λ l y l z l .

We consider the H-valued random vectors Y n := (φ l (X n )) _l∈N . Then {Y n , n ≥ 1} is a sequence of H-valued random vectors and

(3) V n = 1 b ² _n

n

X

ij=1

∞

X

l=1

λ l φ l (X i )φ l (X j ) =

∞

X

l=1

λ l

 1 b n

n

X

k=1

φ l (X k )  ²

=

1 b n

n

X

k=1

Y n

2

. Now, let {X _n , n ≥ 1} be a sequence of real-valued pairwise independent random variables. Then {Y _n , n ≥ 1} is a sequence of H-valued coordinatewise pairwise NQD random vectors. Thus, since the limit result of H-valued coordinatewise pairwise NQD random vectors {Y _n , n ≥ 1}, we obtain the convergence of V _n .

The organization of the paper is as follows. The concept of coordinatewise pairwise NQD random vectors in Hilbert spaces is defined in Section 2, where we also prove some useful lemmas. The results for weak laws of large numbers for weighted coordinatewise pairwise NQD random vectors in Hilbert spaces in the case that the decay order of tail probability is r for some 0 < r < 2 are contained in Section 3. As applications, in Section 4, we give the extended Pareto-Zipf distribution, the generalized St. Petersburg game and present some results about weak laws of large numbers for each model, respectively.

2. Preliminaries

Let {a n ; n ≥ 1} and {b n ; n ≥ 1} be sequences of positive real numbers.

We use notation a n  b n instead of 0 < lim inf a n /b n ≤ lim sup a n /b n < ∞;

a _n = o(b _n ) means that lim _n→∞ a _n /b _n = 0; notation a _n ∼ b _n is used for lim n→∞ a n /b n = 1. These notations are also used for positive real functions f (x) and g(x). The indicator function of A is denoted by I(A). Throughout this paper, the symbol C will denote a generic constant (0 < C < ∞) which is not necessarily the same one in each appearance.

Let H be a real separable Hilbert space with the norm k · k generated by an inner product h·, ·i and let {e j , j ∈ B} be an orthonormal basis in H.

Huan et al. [8] introduced the concept of coordinatewise negatively association (CNA) random vectors in H and Mi-Hwa Ko [10] introduced the concept of coordinatewise asymptotical negatively association (CANA) random vectors in H. In this paper, we present the concept of coordinatewise pairwise NQD random vectors in H.

Definition. A sequence {X n , n ≥ 1} of H-valued random vectors is said to be coordinatewise pairwise NQD if for any j ∈ B, the sequence of random variables {< X n , e j >, n ≥ 1} is pairwise NQD.

Remark 2.1. If a sequence of H-valued random vectors is NA [11] (or CNA, or pairwise independent), then it is coordinatewise pairwise NQD.

Example 2.2. Let {Y n ; n ≥ 1} be a sequence of pairwise NQD random vari- ables. For each n ≥ 1, j ∈ B, put X _n ^j = |c _j |Y n where P

j∈B c ² _j < ∞. We consider X n = P

j∈B X _n ^j e j , n ≥ 1, then {X n , n ≥ 1} is a sequence of H-valued coordinatewise pairwise NQD random vectors.

The following example shows a sequence of H-valued coordinatewise pairwise NQD random vectors which is not pairwise independent.

Example 2.3. Let {Z n , n ≥ 1} be i.i.d. N (0, 1) random variables. Then {Z n − Z n+1 , n ≥ 1} are identically distributed N (0, 2) random variables. Let F be the N (0, 2) distribution function and {F n , n ≥ 1} be a sequence of continuous distribution functions. For n ≥ 1, put

F _n ⁻¹ (t) = inf{x : F _n (x) ≥ t} and Y _n = F _n ⁻¹ (F (Z _n − Z n+1 )).

Li et al. [13] showed that {Y n , n ≥ 1} is a sequence of pairwise NQD random variables and for all n ≥ 1, the distribution function of Y n is F n . For each n ≥ 1, j ∈ B, put X _n ^j = |c j |Y n where P

j∈B c ² _j < ∞. We consider X n = P

j∈B X _n ^j e _j , n ≥ 1 (see Example 2.2), then {X _n , n ≥ 1} is a sequence of H-valued coordinatewise pairwise NQD random vectors.

But

Cov(Z n − Z n+1 , Z n+1 − Z n+2 ) = −1

then X n and X n+1 are not independent. Consequently, {X n , n ≥ 1} is not a sequence of pairwise independent random vectors.

The following lemma plays an essential role in our main results.

Lemma 2.4 ([12]). Let X and Y be R-valued NQD random variables. Then, i) cov(X, Y ) ≤ 0.

ii) If f and g are Borel functions, both of which are monotone increasing (or both are monotone decreasing), then f (X) and g(X) are NQD.

Lemma 2.5. Let (X _n , n ≥ 1) be a sequence of H-valued coordinatewise pair- wise NQD random vectors with mean 0 and finite second moments. Then,

E

n

X

k=1

X k

2

≤

n

X

k=1

EkX k k ² . Proof. For n ≥ 1, we have by Lemma 2.4 that

E

n

X

k=1

X k

2

= E X

j∈B

 h

n

X

k=1

X k , e j i  ²

= X

j∈B

E  X ⁿ

k=1

hX k , e j i  ²

= X

j∈B

 X ⁿ

k=1

EhX k , e j i ² + X

k6=i

cov(hX k , e j i, hX i , e j i) 

≤

n

X

k=1

X

j∈B

EhX k , e j i ² =

n

X

k=1

EkX k k ² .

 Lemma 2.6. Let X be an H-valued random vector. Suppose that

X

j∈B

P (|hX, e _j i| > x)  x ^−r for a fixed 0 < r < 2.

Then (a) P

j∈B

E(|hX, e _j i|I(|hX, e _j i| > x))  x ^1−r if 1 < r < 2.

(b) P

j∈B

E(|hX, e j i| ^α I(|hX, e j i| ≤ x))  x ^α−r if α > r > 0.

Proof. To prove (a), we have X

j∈B

E(|hX, e _j i|I(|hX, e _j i| > x))  X

j∈B

X

k≥x

kP (k < |hX, e _j i| ≤ k + 1)

= X

j∈B

X

k≥x

P (|hX, e _j i| > k)

= X

k≥x

X

j∈B

P (|hX, e _j i| > k)

 X

k≥x

k ^−r  x ^1−r .

For the proof of (b), we see X

j∈B

E(|hX, e j i| ^α I(|hX, e j i| ≤ x))

 X

j∈B

X

0≤k≤x−1

(k + 1) ^α P (k < |hX, e j i| ≤ k + 1)

 X

j∈B

X

0≤k≤x−1

((k + 1) ^α − k ^α )P (|hX, e j i| > k)

 X

j∈B

X

1≤k≤x−1

k ^α−1 P (|hX, e j i| > k)

= X

1≤k≤x−1

k ^α−1 X

j∈B

P (|hX, e j i| > k)

 X

1≤k≤x−1

k ^α−1−r  x ^α−r .

 3. The main results

We consider {X _n , n ≥ 1} is a sequence of coordinatewise pairwise NQD random vectors in H. For k ≥ 1, j ∈ B, we set

X _k ^j = hX k , e j i.

Now, the main results can be stated and proved.

Theorem 3.1. Let {X n ; n ≥ 1} be a sequence of H-valued coordinatewise pairwise NQD random vectors with zero mean and infinite rth moments for some 1 < r < 2 whose distributions satisfy that P

j∈B P (|X _n ^j | > x)  x ^−r for all n ≥ 1 and lim sup _x→∞ sup

n≥1

x ^r P

j∈B P (|X _n ^j | > x) < ∞. Let (a nk ; 1 ≤ k ≤ m n , n ≥ 1) be an array of positive real numbers such that

(4)

m

_n

X

k=1

a ^r _nk = o(1).

Then

m

n

X

k=1

a _nk X _k − ^P → 0 as n → ∞.

Proof. For each j ∈ B, n ≥ 1 and 1 ≤ k ≤ m n . Put

Y _nk ^j = X _k ^j I(|X _k ^j | ≤ a ⁻¹ _kn ) + a nk I(X _k ^j > a ⁻¹ _nk ) − a nk I(X _k ^j < −a ⁻¹ _nk ),

Y _nk = X

j∈B

Y _nk ^j e _j , U _n =

m

n

X

k=1

a _nk [Y _nk − EY nk ].

By using Lemma 2.4, it is easy to see that {Y _nk − EY _nk , 1 ≤ k ≤ m _n } is a sequence of H-valued coordinatewise pairwise NQD random vectors with mean 0. For an arbitrary ε > 0, by Lemma 2.5 and Lemma 2.6, we have

P (kU _n k > ε) ≤ 1

ε ² E(kU _n k ² )

≤ 1 ε ²

m

n

X

k=1

a ² _nk EkY nk − EY nk k ²

= 1 ε ²

m

_n

X

k=1

a ² _nk X

j∈B

E(Y _nk ^j − EY _nk ^j ) ²

≤ 1 ε ²

m

_n

X

k=1

a ² _nk X

j∈B

E(Y _nk ^j ) ²

≤ 1 ε ²

m

_n

X

k=1

a ² _nk X

j∈B

E(|X _k ^j | ² I(|a _nk X _k ^j | ≤ 1))

+ 1 ε ²

m

n

X

k=1

X

j∈B

P (|X _k ^j | > a ⁻¹ _nk )

≤ C ε ²

m

n

X

k=1

a ^r _nk → 0 as n → ∞.

(5)

On the other hand, we also have by Lemma 2.6 that

m

n

X

k=1

a nk kEY nk k =

m

n

X

k=1

a nk

X

j∈B

EY _nk ^j e j

≤

m

n

X

k=1

a _nk

X

j∈B

E 

X _k ^j I(|X _k ^j | ≤ a ⁻¹ _nk )  e _j

+

m

n

X

k=1

X

j∈B

P (|X _k ^j | > a ⁻¹ _nk )

=

m

n

X

k=1

a _nk

X

j∈B

E 

X _k ^j I(|X _k ^j | > a ⁻¹ _nk )  e _j

+

m

n

X

k=1

X

j∈B

P (|X _k ^j | > a ⁻¹ _nk )

≤

m

n

X

k=1

a _nk X

j∈B

E 

|X _k ^j |I(|X _k ^j | > a ⁻¹ _nk ) 

+

m

_n

X

k=1

X

j∈B

P (|X _k ^j | > a ⁻¹ _nk )

≤ C

m

n

X

k=1

a ^r _nk → 0 as n → ∞.

(6)

Consequently, (5) and (6) yield that P m

_n

k=1 a _nk Y _nk converges in probability to 0. Finally, put V _n = P n

k=1 a _nk Y _nk and S _n = P m

_n

k=1 a _nk X _k , we have that for arbitrary  > 0,

P (kV _n − S n k > ) ≤ X

k≤m

n

P (Y _nk 6= X k ) ≤ X

k≤m

n

X

j∈B

P (Y _nk ^j 6= X _k ^j )

= X

k≤m

n

X

j∈B

P (|X _k ^j | > a ⁻¹ _nk ) ≤ C

m

_n

X

k=1

a ^r _nk → 0 as n → ∞.

Therefore, S n

→ 0 as n → ∞. P 

When m n = n and a nk = _b ¹

n

(1 ≤ k ≤ n), we obtain the following corollary.

Corollary 3.2. Let {X n ; n ≥ 1} be a sequence of H-valued coordinatewise pairwise NQD random vectors with zero mean and infinite rth moments for some 1 < r < 2 whose distributions satisfy that P

j∈B P (|X _n ^j | > x)  x ^−r for n ≥ 1 and lim sup _x→∞ sup _n≥1 x ^r P

j∈B P (|X _n ^j | > x) < ∞. Let {b n ; n ≥ 1} be a sequence of positive real numbers such that

n→∞ lim n ^1/r

b _n = 0.

Then,

1 b n

n

X

k=1

X _k − ^P → 0 as n → ∞.

Example 3.3. Let {X n , n ≥ 1} be the sequence of random vectors defined in Example 2.3 with (c j ) satisfying the condition P

j∈B |c j | ^r < ∞ for some 1 < r < 2 and {F _n , n ≥ 1} be the distribution functions of a common density function

f (x) =

( r

2x ^r+1 for |x| > 1, 0 otherwise.

One readily checks that, EX ₁ = 0, EkX ₁ k ^r = ∞ so that the Marcinkiewicz- Zygmund weak law does not hold. However, it is easy to see that P

j∈B P (|X ₁ ^j |

> x)  x ^−r then applying Corollary 3.2 we get 1

n ^1/r l(n)

n

X

k=1

X k

− P → 0 as n → ∞,

where lim n→∞ l(n) = ∞.

The obvious question that comes to mind is whether or not there is almost sure convergence in Corollary 3.2. The following example shows that the weak law established in Corollary 3.2 cannot be extended to a strong law.

Example 3.4. Consider the real Hilbert space ` ₂ of all square summable real sequences with inner product

hx, yi =

∞

X

i=1

x _i y _i for x, y ∈ ` ₂ .

The standard orthonormal basis of ` <sub>2</sub> is {e <sub>n</sub> , n ≤ 1} where e <sub>n</sub> denote the element of ` ₂ having 1 in its nth position and 0 elsewhere. For each 1 < r ≤ 2, let {X _n , n ≥ 1} be a sequence of i.i.d. symmetric random vectors in ` ₂ space such that

X n = (X _n ¹ , X _n ² , . . . , X _n ^j , . . . ), where

P (|X _n ^j | > x) = c j x ^−r (c j > 0,

∞

X

j=1

c j < ∞) for all x > x 0 > 0, and j = 1, 2, . . . .

Then the hypotheses of Corollary 3.2 are met. Now, let b _n = (n log n) ^1/r , n ≥ 1.

For each j > 0, we obtain

∞

X

n=1

P (|X _n ^j | > cb n )  c

∞

X

n=1

1 b ^r _n = c

∞

X

n=1

1

n log n = ∞ for any c > 0, so that, by the Borel-Cantelli lemma,

P (|X _n ^j | > cb n infinitely often) = 1 for any c > 0.

Thus

lim sup

n→∞

X _n ^j b n

= ∞ almost surely.

On the other hand, X _n ^j

b n

= P n

k=1 X _k ^j b n

− b n−1

b n

P n−1 k=1 X _k ^j b n−1

. This implies for each j = 1, 2, . . . ,

P n k=1 X _k ^j

b n

6→ 0 almost surely.

Therefore, SLLN

P n k=1 X _k

b n

→ 0 almost surely

fails.

Theorem 3.5. Let {X _n ; n ≥ 1} be a sequence of H-valued coordinatewise pairwise NQD random vectors whose distributions satisfying P

j∈B P (|X _n ^j | >

x)  x ^−r for n ≥ 1 and lim sup _x→∞ sup _n≥1 x ^r P

j∈B P (|X _n ^j | > x) < ∞ (0 <

r ≤ 1). Let (a nk ; 1 ≤ k ≤ m n , n ≥ 1) be an array of positive real numbers such that

(7)

m

_n

X

k=1

a ^r _nk = o(1).

Then (8)

m

_n

X

k=1

a _nk 

X _k − EY _nk  _P

− → 0 as n → ∞,

where Y nk are defined as in proof of Theorem 3.1.

Proof. It is well known that {Y nk − EY nk , 1 ≤ k ≤ m n } is a sequence of coordinatewise pairwise NQD random vectors in H. Let ε be an arbitrary positive number. We have, by Lemma 2.6,

P 

m

n

X

k=1

a _nk 

Y _nk − EY nk



> ε 

≤ 1 ε ² E

m

_n

X

k=1

a nk



Y nk − EY nk



2

≤ 1

 ²

m

_n

X

k=1

a ² _nk EkY nk − EY nk k ² = 1

 ²

m

_n

X

k=1

a ² _nk X

j∈B

E(Y _nk ^j − EY _nk ^j ) ²

≤ 1

 ²

m

_n

X

k=1

a ² _nk X

j∈B

E(Y _nk ^j ) ²

≤ 1

 ²

m

n

X

k=1

a ² _nk X

j∈B

E(|X _k ^j | ² I(|a _nk X _k ^j | ≤ 1)) + 1

 ²

m

n

X

k=1

X

j∈B

P (|X _k ^j | > a ⁻¹ _nk )

≤ C

m

n

X

k=1

a ^r _nk → 0 as n → ∞.

This implies that (9)

m

n

X

k=1

a nk



Y nk − EY nk

 _P

− → 0 as n → ∞.

On the other hand, we see P (

m

n

X

k=1

a _nk (X _k − Y nk )

> ) ≤ X

k≤m

n

P (Y _nk 6= X k ) ≤ X

k≤m

n

X

j∈B

P (Y _nk ^j 6= X _k ^j )

= X

k≤m

_n

X

j∈B

P (|X _k ^j | > a ⁻¹ _nk )

≤ C

m

_n

X

k=1

a ^r _nk → 0 as n → ∞.

(10)

(9) and (10) yield the conclusion (8). 

Corollary 3.6. Under the assumptions of Theorem 3.5, if 0 < r < 1, then we have

(11)

m

_n

X

k=1

a _nk X _k − ^P → 0 as n → ∞.

Proof. By Lemma 2.6, we have

m

_n

X

k=1

a nk EY nk

≤

m

_n

X

k=1

a nk kEY nk k ≤

m

_n

X

k=1

a nk

X

j∈B

E|Y _nk ^j |

≤

m

n

X

k=1

a _nk X

j∈B

E 

|X _k ^j |I(|X _k ^j | ≤ a ⁻¹ _nk ) 

+

m

n

X

k=1

X

j∈B

P (|X _k ^j | > a ⁻¹ _nk )

≤ C

m

_n

X

k=1

a ^r _nk → 0 as n → ∞.

(12)

 Example 3.7. Let {X n , n ≥ 1} be the sequence of random vectors defined in Example 2.3 with (c j ) such that P

j∈B |c j | ^r < ∞ for some 0 < r < 1 and {F n , n ≥ 1} be the distribution functions of common density function

f (x) =

( r

2x ^r+1 for |x| > 1, 0 otherwise.

Then P

j∈B P (|X ₁ ^j | > x)  x ^−r . Using Corollary 3.6 with a nk = _n

_1/r

¹ _l(n) , 1 ≤ k ≤ n, where lim n→∞ l(n) = ∞, we get

1 n ^1/r l(n)

n

X

k=1

X k

− P → 0 as n → ∞.

In the case l(n) = (log n) ^1/r , we get the same result as in Example 6.4.4 (p. 282) of [7].

Corollary 3.8. Under the assumptions of Theorem 3.5, if r = 1 and there exists A ∈ H such that

lim

n→∞

m

n

X

k=1

a _nk X

j∈B

EX _k ^j I(|X _k ^j | ≤ a ⁻¹ _nk )e _j = A,

(13)

then we have (14)

m

_n

X

k=1

a nk X k

− P → A as n → ∞.

Proof. We have

m

_n

X

k=1

a _nk EY _nk −

m

_n

X

k=1

a _nk X

j∈B

EX _k ^j I(|X _k ^j | ≤ a ⁻¹ _nk )e _j

m

n

X

k=1

a _nk X

j∈B

E|Y _nk ^j − X _k ^j I(|X _k ^j | ≤ a ⁻¹ _nk )|

=

m

n

X

k=1

X

j∈B

P (|X _k ^j | > a ⁻¹ _nk ) ≤ C

m

n

X

k=1

a ^r _nk → 0 as n → ∞.

Then

n→∞ lim

m

n

X

k=1

a _nk EY _kn = A,

by Theorem 3.5, we obtain (14). 

In the following corollary, we restrict to the identically distributed coordi- natewise pairwise NQD random vector case. We recall the concept of slowly varying function at infinity as follows: Let a ≥ 0, a positive measurable function f (x) on [a; ∞) is said to be slowly varying at infinity if

x→∞ lim f (tx)

f (x) = 1 for all t > 0.

Clearly, log x, log log x are slowly varying functions at infinity (the readers may refer to [5, 7]).

Corollary 3.9. Let {X _n ; n ≥ 1} be a sequence of H-valued identically dis- tributed coordinatewise pairwise NQD random vectors whose common distri- butions satisfy P

j∈B P (|X ^j | > x)  x ⁻¹ . In addition, we suppose that there exists a sequence of real numbers (α _j ) _j∈B such that P

j∈B α _j ² < ∞ and

x→∞ lim

EX ^j I(|X ^j | ≤ x)

log x = α j (j ∈ B) (15)

then for each real number β > −1 and a slowly varying sequence l(n) we have

(16) 1

n ^β+1 l(n) log n

n

X

k=1

k ^β l(k)X k

− P → α

1 + β as n → ∞, where α = P

j∈B α _j e _j .

Proof. Using Corollary 3.8, with m _n = n, a _nk = _c ^b

^k

n

, b _k = k ^β l(k) and c _n = n ^β+1 l(n) log n. We note

n

X

j=1

b _j ∼ 1

1 + β n ^β+1 l(n),

n

X

j=1

b _j log b _j ∼ β

1 + β n ^β+1 l(n) log n and log c _n ∼ (β + 1) log n,

X

j∈B

E(|X ^j |I(|X ^j | ≤ x))  X

j∈B

X

0≤k≤x−1

(k + 1)P (k < |X ^j | ≤ k + 1)

= X

j∈B

X

1≤k≤x−1

P (|X ^j | > k)

= X

1≤k≤x−1

X

j∈B

P (|X ^j | > k)

 X

1≤k≤x−1

k ⁻¹  log x.

This implies that X

j∈B

  

n

X

k=1

a _nk EX ^j I(|X ^j | ≤ a ⁻¹ _nk )    ≤ 1

c n n

X

k=1

b _k X

j∈B

E|X ^j |I(|X ^j | ≤ c _n b k

)

 1 c n

n

X

k=1

b _k log( c _n b k

) → 1

1 + β as n → ∞, so

(17) sup

n

X

j∈B

  

n

X

k=1

a nk EX ^j I(|X ^j | ≤ a ⁻¹ _nk )    < ∞.

Moreover, for each j ∈ B, by (15) we have

n

X

k=1

a nk EX ^j I{|X ^j | ≤ a ⁻¹ _nk } ∼ α j

P n

k=1 b _k log( ^c _b

ⁿ

k

)

c _n → α j

1 + β as n → ∞.

Using (17),

A = lim

n→∞

n

X

k=1

a nk

X

j∈B

EX _k ^j I(|X _k ^j | ≤ a ⁻¹ _nk )e j → α

1 + β as n → ∞.

This completes the proof. 

4. Applications 4.1. Extended Pareto-Zipf distribution

In this subsection, we extend Theorem 3.1 of Nakata [16] as follows.

We consider {X n , n ≥ 1} as a sequence of coordinatewise pairwise NQD random vectors in H whose distributions are defined by P (X _n ^j = 0) = 1 − ^α _c

r^j

n

, (j ∈ B) and the tail probability

P (X _n ^j > x) = α _j

(x + c n ) ^r for x > 0, j ∈ B, where r > 0, α _j ≥ 0, P

j∈B α _j < ∞ and {c _n ; n ≥ 1} is a sequence of positive numbers such that

c ^r _n ≥ max{α _j } for all n.

Clearly,

X

j∈B

P (|X _n ^j | > x)  x ^−r for n ≥ 1 and

lim sup

x→∞

sup

n≥1

x ^r X

j∈B

P (|X _n ^j | > x) < ∞.

We have the following theorem.

Theorem 4.1. Let 0 < r < 2; r 6= 1. Suppose that C n =

n

X

k=1

c ^−r _k → ∞ as n → ∞, and

C n = o(b ^r _n ).

If 0 < r < 1, then

(18) 1

b n n

X

k=1

X k

c k

− P → 0 as n → ∞.

If 1 < r < 2, then

(19) 1

b n

 X ⁿ

k=1

X _k c k

− C _n α r − 1

 _P

− → 0 as n → ∞, where α = P

j α j e j .

Proof. If 0 < r < 1, it is easy to obtain (18) by Corollary 3.6.

In the case 1 < r < 2, we have by Theorem 3.1 that 1

b n n

X

k=1

1 c k

(X _k − E(X k )) − ^P → 0 as n → ∞.

Moreover, 1 b _n

n

X

k=1

1

c _k EX k = 1 b _n

n

X

k=1

1 c _k

X

j∈B

EX _k ^j e ^j

= 1 b _n

n

X

k=1

1 c _k

X

j∈B

( Z ∞

0

P (X _k ^j > x)dx)e ^j

= 1 b _n

n

X

k=1

1 c _k

X

j∈B

( Z ∞

0

α j

(x + c _k ) ^r dx)e ^j

= α b _n

n

X

k=1

1

(r − 1)c ^r _k = C n α (r − 1)b _n .

This completes the proof. 

Theorem 4.2. Let r = 1. Suppose that C n =

n

X

k=1

c ⁻¹ _k → ∞ as n → ∞.

Then,

(20) 1

C n log C n n

X

k=1

X k

c k

− P → α as n → ∞.

Proof. From Corollary 3.8, with a nk = _c ¹

k

C

_n

log C

_n

(1 ≤ k ≤ n), it is sufficient to show A = α. It follows that for large x > 0,

X

j∈B

EX _k ^j I(X _k ^j ≤ x)e _j = X

j∈B



xP (X _k ^j ≤ x) − Z x

0

P (X _k ^j ≤ t)dt 

e _j ∼ α log x c _k , then

A = lim

n→∞

1 C _n log C _n

n

X

k=1

1 c _k

X

j∈B

EX _k ^j I(X _k ^j ≤ C n log C n c k )e j

= α lim

n→∞

C n log(C n log C n ) C n log C n

= α.

 4.2. A generalized St. Petersburg Game

We consider {X n , n ≥ 1} as a sequence of identically distributed coordinate- wise pairwise NQD random vectors in H with

P (X _n ^j = 2 ^k/r ) = α j 2 ^−k for k ≥ 1, n ≥ 1, j ∈ B, and P (X _n ^j = 0) = 1 − α j , where 0 < r < 2, 0 ≤ α _j ≤ 1, P

j∈B α _j < ∞. Put α = P

j∈B α _j e _j .

If 1 < r < 2, then m _j = EX _n ^j = ₁₋₂ ^α

_1/r−1^j

, m = P

j m _j e _j = E(X _n ) =

α

1−2

^1/r−1

. Thus, we also have P

j∈B P (|X _n ^j − m j | > x)  x ^−r . Therefore, applying Corollary 3.2 with b n = n ^1/r log log n, we get

1 n ^1/r log log n

n

X

k=1

(X k − m) − ^P → 0 as n → ∞.

If 0 < r < 1, we also have P

j∈B P (|X _n ^j | > x)  x ^−r . Therefore, applying Corollary 3.6 with a _nk = _n

_1/r

_{log log n} ¹ (1 ≤ k ≤ n), we get

1 n ^1/r log log n

n

X

k=1

X k

− P → 0 as n → ∞.

If r = 1, note that EX ^j I _{|X

j

|≤x} ∼ α _j log x/ log 2 for all j ∈ B. By Corollary 3.9, for β > −1 and a slowly varying sequence l(n), we have

n→∞ lim P n

j=1 j ^β l(j)X j

n ^β+1 l(n) log n = 1

(1 + β) log 2 in probability.

Acknowledgments. The authors would like to express their deep thanks to anonymous referees for their useful comments.

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Dung Van Le

Department of Mathematics

The University of Da Nang - University of Science and Education 459 Ton Duc Thang, Da Nang, Vietnam

Email address: [email protected]

Son Cong Ta

Faculty of Mathematics Mechanics and Informatics VNU Hanoi University of Science 334 NguyenTrai, Hanoi, Vietnam Email address: [email protected]

Cuong Manh Tran

Faculty of Mathematics

Mechanics and Informatics

VNU Hanoi University of Science

334 NguyenTrai, Hanoi, Vietnam

Email address: [email protected]