Strong laws of large numbers for weighted sums of negatively dependent random variables

STRONG LAWS OF LARGE NUMBERS FOR WEIGHTED SUMS OF NEGATIVELY

DEPENDENT RANDOM VARIABLES

Mi-Hwa Ko, Kwang-Hee Han, and Tae-Sung Kim

Abstract. For double arrays of constants {a

, 1 ≤ i ≤ k

, n ≥ 1} and sequences of negatively orthant dependent random variables {X

, n ≥ 1}, the conditions for strong law of large number of P

i=1

a

X

are given. Both cases k

↑ ∞ and k

= ∞ are treated.

1. Introduction

The history and literature on strong laws of large numbers is vast and rich as this concept is crucial in probability and statistical the- ory. The literature on concepts of negative dependence is much more limited but still very interesting ([3], [4] and [7]). Negative dependence has been particularly useful in obtaining strong laws of large numbers (cf. [1], [5], [6], [8] and [10]).

Lehmann [6] provided the concept of negative dependence in the bi- variate case as follows:

Random variables X and Y are negatively quadrant dependent(NQD) if

(1) P {X ≤ x, Y ≤ y} ≤ P {X ≤ x}P {Y ≤ y}

for all x, y ∈ R. A collection of random variables {X _n , n ≥ 1} is said to be pairwise NQD if every pair of random variables in the collection

Received August 12, 2005.

2000 Mathematics Subject Classification: 60F15.

Key words and phrases: negatively quadrant dependent, negatively orthant de- pendent, strong law of large number, weighted sum, double array, stochastically dominated.

This work was partially supported by the Korean Science and Engineering Foun-

dation (R01-2005-000-10696-0) and the SRC/ERC program of MOST/KOSEF (R11-

2000-073-00000).

satisfies (1). It is important to note that (1) implies (2) P {X > x, Y > y} ≤ P {X > x}P {Y > y}

for all x, y ∈ R. Moreover, it follows that (2) implies (1), and hence, they are equivalent for pairwise NQD.

Ebrahimi and Ghosh [2] showed that (1) and (2) are not equivalent for n ≥ 3(i.e., (3) and (4) below are not equivalent). Consequently, the following definition is needed to define sequences of negatively dependent random variables:

The random variables X ₁ , X ₂ , . . . , X _n are said to be

(a) lower negatively orthant dependent(LNOD) if for each n (3) P {X ₁ ≤ x ₁ , . . . , X _n ≤ x _n } ≤

Y n i=1

P {X _i ≤ x _i } for all x ₁ , . . . , x _n ∈ R,

(b) upper negatively orthant dependent(UNOD) if for each n (4) P {X ₁ > x ₁ , . . . , X _n > x _n } ≤

Y n i=1

P {X _i > x _i } for all x ₁ , . . . , x _n ∈ R,

(c) negatively orthant dependent (NOD) if both (3) and (4) hold.

This notion was introduced by Ebrahimi and Ghosh [2]. In this paper we will give some results of almost sure convergence of weighted sums of NOD sequences, which have not appeared before. In section 2 we consider the case of triangular-type weight arrays and in section 3 we consider the case of infinite double arrays.

We will use the following concept in this paper. Let {X _n , n ≥ 1}

be a sequence of random variables and let X be a nonnegative ran- dom variable. If there exists a constant C(0 < C < ∞) satisfying sup _n≥1 P {|X _n | ≥ t} ≤ CP {X ≥ t} for any t ≥ 0, then {X _n , n ≥ 1} is said to be stochastically dominated by X (briefly {X _n , n ≥ 1} ≺ X).

Throughout the remainder of this paper, C will stand for a constant whose value may vary from line to line.

2. Triangular arrays

Lemma 2.1. ([8]) If {X _n , n ≥ 1} is a sequence of NOD random vari-

ables and {f _n } is a sequence of Borel functions all of which are monotone

increasing (or all monotone decreasing), then {f _n (X _n )} is a sequence of

NOD random variables.

Theorem 2.2. ([8]) Let {X _i , 1 ≤ i ≤ n} be a sequence of nonnegative random variables which are upper negatively orthant dependent. Then (5) E(Π ⁿ _i=1 X _i ) ≤ Π ⁿ _i=1 E(X _i ).

Theorem 2.3. Let 0 < r ≤ 2. Assume that {X _n , n ≥ 1} is a sequence of NOD random variables and stochastically dominated by a nonnegative random variable X, i.e., X _n ≺ X. Let {a _ni , 1 ≤ i < k _n < ∞, 1 ≤ k _n ↑, n ≥ 1} be an array of constants and {d _n , n ≥ 1} a sequence of constants such that 0 < d _n ↑ and d _n = O(n

). If the following conditions are satisfied:

(i) there exists a positive number p such that (6)

X ∞ n=1

k ^−p _n < ∞,

(7) (ii)

X ∞ n=1

P (X ≥ d _n ) < ∞,

(8) (iii) max

1≤i≤k

|a _ni | d _i = O( 1 log k _n ) and

(iv) one of the following statements is satisfied a) if 1 ≤ r ≤ 2, ^d _n

↓, EX _n = 0,

(9)

k

X

i=1

a ² _ni d ^2−r _i = o( 1 log k _n ) b) if 0 < r < 1,

(10)

k

X

i=1

|a _ni | d ^1−r _i = o(1).

Then (11)

k

X

i=1

a _ni X _i → 0 a.s.

Proof. In view of d _n = O(n

), (7) yields EX ^r < ∞. From (7)

and {X _n } ≺ X, Σ ^∞ _n=1 P {|X _n | > d _n } < ∞, i.e., P {|X _n | > d _n , i.o.} = 0

follows. In the following statement, without loss of generality we assume

a _ni ≥ 0, n ≥ 1, i ≥ 1 and EX ^r = 1. If q _n = max _1≤i≤k

|a _ni |d _i . Then q _n

= O( _{log k} ¹

) by (8).

(a) If 1 ≤ r ≤ 2, let S _n

=

k

X

i=1

a _ni (X _i

− EX _i

) and S _n

=

k

X

i=1

a _ni (X _i

− EX _i

), where X _i

= (−d _i ) ∨ (X _i ∧ d _i ) and X _i

= X _i − X _i

. Obviously,

S _n =

k

X

n=1

a _ni X _i = S _n

+ S _n

.

Now if g(x) = x ⁻² (e ^x −1−x), then g(x) is nonnegative and increasing ([9]). For t > 0 and 1 ≤ i ≤ k _n , noting that|X _i

−EX _i

| ≤ 2d _i and E|X _i

− EX _i

| ^r ≤ 2 ^r−1 (2E|X _i | ^r ) ≤ C2 ^r , where C comes from sup _n P (|X _n | ≥ t) ≤ CP (X ≥ t), we have

E exp{ta _ni (X _i

− EX _i

)}

= 1 + E{exp[ta _ni (X _i

− EX _i

)] − 1 − ta _ni (X _i

− EX _i

)}

≤ 1 + t ² a ² _ni g(2tq _n )E{|X _i

− EX _i

| ^r (2d _i ) ^2−r }

≤ exp{4Ct ² a ² _ni d ^2−r _i g(2tq _n )}.

(12)

From (12) and Theorem 2.2 , it follows that (13) E(e ^tS

) ≤ exp{4Ct ² g(2tq _n )

k

X

i=1

a ² _ni d ^2−r _i }.

For all ² > 0, set t = (p + 1)ε ⁻¹ log k _n . From (8) and (9), there exists a positive constant K _ε such that

P (S _n

> ε)

≤ e ^−εt Ee ^tS

≤ exp{−(p + 1) log k _n

+4C(p + 1) ² ε ⁻² g(K _ε )(log k _n ) ² (

k

X

i=1

a ² _ni d ^2−r _i )}

≤ exp{−p log k _n } = k ^−p _n . (14)

From (6) and (13), it follows that

(15) lim _n→∞ S _n

≤ 0 a.s.

Replacing X _i by − X _i , we get

(16) lim _n→∞ (−S _n

) ≤ 0. a.s.

Consequently, by (15) and (16)

(17) S _n

→ 0 a.s.

Now we will show that

(18) S _n

→ 0 a.s.

If there exists 0 < K < ∞ such that d _n < K, n ≥ 1, then from (7) and {X _n } ≺ X, we have X ≤ K a.s. for all n ≥ 1. Replacing d _n by K we get S _n

= 0 a.s. and, of course, S _n

→ 0 a.s.

Now we assume that d _n ↑ ∞. On the other hand, from the condi- tions about d _n there exists a function d(x) on (0, ∞) which is positive, continuous, increasing, d(x)/x ↓ and d(n) = d _n . Set d ⁻ (x) = inf{y >

0, d(y) ≥ x}. It is easy to know that d ⁻ (x)/x ↑ on (0, ∞) and d ⁻ (X) is integrable. Thus we have

n d _n

Z

{|X

|≥d

}

|X _n |dp < C n d _n

Z

{X≥d

}

Xdp

≤ C Z

{X≥d

}

d ⁻ (X)dp = o(1).

So there exist a sequence of positive constant o(1) and C > 0 such that (

k

X

i=1

a _ni EX _i

) ² ≤ (

k

X

i=1

a _ni ² _i d _i i ⁻¹ ) ²

≤ C(

k

X

i=1

a ² _ni d ^2−r _i )(

k

X

i=1

² ² _i i ⁻¹ )

= o(1).

(19)

From (9), noting that d _n ↑, there exists C > 0 such that (20)

k

X

i=1

a ² _ni ≤ C(

k

X

i=1

a ² _ni d ^2−r _i ) = o( 1 log k _n ).

From (20), we obtain (

k

X

i=1

a _ni X _i

) ² ≤ (

k

X

i=1

a ² _ni )(

k

X

i=1

|X _i | ² I(|X _i | ≥ d _i ) (21)

From (19) and (21), S _n

→ 0 a.s., i.e., (11) holds.

(b) If 0 < r < 1, let S

_n = P _k

i=1 a _ni X _i

and S _n

= P _k

i=1 a _ni X _i

, where X _i

= (−d _i ) ∨ (X _i ∧ d _i ) and X _i

= X _i − X _i

. Obviously, P _k

i=1 a _ni X _i = S _n

+ S _n

. Set g ₁ (x) = x ⁻¹ (e ^x − 1), then g ₁ (x) is an increasing function.

Since E|X _i | ^r ≤ CEX ^r = C (where C comes from sup _n P (|X _n | ≥ t) ≤ CP (|X| ≥ t)), we have

Eexp(ta _ni X _i

) ≤ 1 + Cta _ni d ^1−r _i g ₁ (tq _n )

≤ exp{Cta _ni d ^1−r _i g ₁ (tq _n )}

and

(22) Eexp(tS

_n ) ≤ exp{Ctg ₁ (tq _n )

k

X

i=1

a _ni d ^1−r _i }

by Theorem 2.2. Let t = (p + 1)ε ⁻¹ log k _n . It follows as earlier method that

S _n

→ 0 a.s.

On the other hand, via (10), there exists C > 0 such that

k

X

i=1

a ² _ni ≤ C(

k

X

i=1

a _ni d ^1−r _i ) ² = o(1).

Consequently,

S _n

² = |

k

X

i=1

a _ni X _i

| ²

≤ (

k

X

i=1

a ² _ni )(

k

X

i=1

|X _i | ² I(|X _i | ≥ d _i )) → 0 a.s.

So (11) holds. The proof is complete.

According to next theorem a slight strengthening of (8) permits a weakening of (9):

Theorem 2.4. Let {X _n , n ≥ 1} be a sequence of N OD random vari- ables and stochastically dominated by a nonnegative random variable X, i.e., X _n < X, and let {a _ni , 1 ≤ i < k _n < ∞, 1 ≤ k _n ↑, n ≥ 1} be an array of constants. If E|X| ^r < ∞, 1 ≤ r ≤ 2, EX = 0 and (6) holds, then

(23) max

1≤i≤k

|a _ni | = O(k ⁻ _n

(log k _n ) ⁻¹ )

implies (11).

Proof. Choosing d _i = i and letting

X _i ⁰ = (−i

) ∨ (X ∧ i

) and X _i ⁰⁰ = X _i − X _i ⁰ ,

the proof follows the same pattern as that of Theorem 2.3 noting that (23)

k

X

i=1

a ² _ni ≤ k _n

(log k _n ) ⁻² .

Remark. If P _k

i=1 a ² _ni i

⁻¹ = o( _{log k} ¹

n

), then (23) can be weakened to

1≤i≤k max

|a _ni |i

= O( 1 log k _n ).

Moreover, if P _k

i=1 a ² _ni d ^2−r _i = o( _{log k} ¹

), where d _n = O(n

) then (23) can be weakened to (8).

Lemma 2.5. Let {X _n , n ≥ 1} be a sequence of NOD random variables with EX _n = 0, sup _n |X _n | ≤ C a.s. (0 < C < ∞). If {a _ni , 1 ≤ i ≤ k _n ↑

∞, n ≥ 1} is a sequence of constants satisfying

(24) (i) max

1≤i≤k

|a _ni | = O( 1 k _n ), (ii) there exists p > 0 such that

X ∞ n=1

1 k ^p _n < ∞, (25)

then X k

i=1

a _ni X _i → 0 a.s.

Proof. Assume a _ni ≥ 0. From the inequality e ^x ≤ 1 + x + ¹ ₂ x ² e ^|x| and (24) it follows that, for all t > 0

Eexp{ta _ni X _i } ≤ 1 + E{ 1

2 t ² a ² _ni X _i ² exp(ta _ni |X _i |)}

≤ exp{C t ²

k _n ² exp(C t k _n )}.

By Theorem 2.2 we get Eexp{t

k

X

i=1

a _ni X _i } ≤ exp{C t ²

k _n exp(C t

k _n )}.

For all ² > 0, set t = p( ^{log k} _²

). Noticing that C _k ^t

exp(C _k ^t

) is bounded, it follows that

X ∞ n=1

P {

k

X

i=1

a _ni X _i > ²}

≤ X ∞ n=1

exp{−t² + C t ²

k _n exp(C t k _n )}

≤ C X ∞ n=1

exp(−p log k _n ) < ∞.

Consequently, we have

n→∞ lim

k

X

i=1

a _ni X _i ≤ 0 a.s.

Replacing X _i by −X _i , we obtain

n→∞ lim

k

X

i=1

a _ni (−X _i ) ≤ 0 a.s.

So

n→∞ lim

k

X

i=1

a _ni X _i = 0 a.s.

The proof is complete.

Theorem 2.6. Let {X, X _n , n ≥ 1} be a sequence of identically dis- tributed NOD random variables with EX = 0 and {a _ni , 1 ≤ i ≤ k _n , ↑

∞, n ≥ 1} an array of constants satisfying max _1≤i≤k

|a _ni | = O( _k ¹

). If there exists a positive p such that P _∞

n=1 1

k

< ∞, then P _k

i=1 a _ni X _i → 0 a.s.

Proof. Assume a _ni ≥ 0 and let X _i

= (−M ) W (X _i V

M ) and X _i

= X _i − X _i

. Then {X _i

− EX _i

} satisfies the conditions in Lemma 2.5 and thus

(26)

k

X

i=1

a _ni (X _i

− EX _i

) → 0 a.s.

On the other hand, since max _1≤i≤k

|a _ni | = O( _k ¹

)

|

k

X

i=1

a _ni (X _i

− EX _i

)|

≤ C

k _n

k

X

i=1

|X _i |I(|X _i | > M ) + CE|X|I(|X| > M ).

Since NOD sequence is pairwise NQD, we get 1

k _n

k

X

i=1

|X _i |I(|X _i | > M ) → E|X|I(|X| > M ) a.s.

by Theorem 1 in [8]. Noticing that E|X|I(|X| > M ) → 0 as M → ∞, we obtain

(27)

k

X

i=1

a _ni (X _i

− EX _i

) → 0 a.s.

From (26) and (27), P _k

i=1 a _ni X _i → 0 a.s.

Theorem 5 in [1] is a special case of the above theorem.

Corollary 2.7. Suppose that {X, X _n , n ≥ 1} is a sequence of iden- tically distributed and independent random variables with EX = 0 and {a _ni , 1 ≤ i ≤ n, n ≥ 1} is a triangular array of constants satisfying max _1≤i≤n |a _ni | = O( _n ¹ ). Then P _n

i=1 a _ni X _i → 0 a.s.

3. Infinite double arrays

Lemma 3.1. Assume that {X _n , n ≥ 1} is a sequence of NOD random variables with EX _n = 0 and |X _n | ≤ d _n , n ≥ 1 and {a _ni , i ≥ 1, n ≥ 1} is an array of constants satisfying

(28) sup

i

d _i |a _ni | = o( 1 log n ).

If one of the following conditions holds (i) EX _n ² ≤ σ ² < ∞, n ≥ 1 and

X ∞ i=1

a ² _ni = o( 1 log n ), (29)

(ii) E|X _n | ≤ α < ∞, n ≥ 1 and X ∞ i=1

|a _ni | = o(1).

(30)

Then

(31) S _n =

X ∞ i=1

a _ni X _i → 0 a.s..

Proof. Assume a _ni ≥ 0. Obviously. E P _∞

i=1 a _ni |X _i | < ∞ thus S _n , n ≥ 1 are almost surely defined. From (28), there exist 0 < γ _n = o(1) such that |a _ni X _i | ≤ d _i a _ni ≤ _{log n} ^γ

, n ≥ 1, i ≥ 1.

If (i) holds, for all t > 0, we can show similarly to the display (12) that

Eexp{ta _ni X _i } = 1 + t ² a ² _ni EX _i ² g(ta _ni X _i )

≤ exp{σ ² t ² a ² _ni g( tγ _n log n )}.

Since {X, X _n , n ≥ 1} are NOD, by applying Fatou’s lemma we obtain Eexp(t

X ∞ i=1

a _ni X _i ) ≤ exp{σ ² t ² g( tγ _n log n )

X ∞ i=1

a ² _ni }.

From (29), there exists 0 < δ _n = o(1) such that P _∞

i=1 a ² _ni = _{log n} ^δ

. Set t _n = ρ ⁻¹ _n log n, where ρ _n = max(δ _n

, γ _n

), necessarily ρ _n = o(1). For all

² > 0 and enough large n, we have P {

X ∞ i=1

a _ni X _i > ²}

≤ exp{−t _n ² + σ ² t ² _n g( t _n γ _n log n )

X ∞ i=1

a ² _ni }

= exp{− ² log n

ρ _n (1 + o(1))} ≤ 1 n ² . So P _∞

n=1 P { P _∞

i=1 a _ni X _i > ²} < ∞, i.e., lim _n→∞ P _∞

i=1 a _ni X _i ≤ 0 a.s..

In the same way we can show that lim _n→∞ P _∞

i=1 a _ni (−X _i ) ≤ 0 a.s..

Thus (31) holds.

If (ii) holds, for all t > 0, we can show similarly to the display pre- ceding (22) that

Eexp{ta _ni X _i } = 1 + ta _ni EX _i g ₁ (ta _ni X _i )

≤ exp{ta _ni EX _i g ₁ (ta _ni X _i )}

≤ exp{αtg ₁ ( tγ _n

log n )a _ni }.

Consequently, Eexp(t

X ∞ i=1

a _ni X _i ) ≤ exp{αtg ₁ ( tγ _n log n )

X ∞ i=1

a _ni }.

Set t _n = ρ ⁻¹ _n log n with ρ _n = γ _n

, then for all ² > 0, P {

X ∞ i=1

a _ni X _i > ²} ≤ 1 n ² From this, (31) holds too.

Theorem 3.2. Suppose that {X _n , n ≥ 1} is a sequence of NOD random variables with EX _n = 0, X is a nonnegative random variable such that {X _n } ≺ X and {a _ni , i ≥ 1, n ≥ 1} is an array constants satisfying

(32)

X ∞ i=1

a ² _ni = o( 1

log n ) and sup

i≥1

|a _ni |i

= o( 1 log n ).

If for some ² > 0,

(33) EX ² (log ⁺ X)

[log ⁺ (log ⁺ X)]

^+ε < ∞

then X ∞

i=1

a _ni X _i → 0 a.s..

Proof. Assume a _ni ≥ 0. Let X _i

= (−i

)∨(X _i ∧i

) and X _i

= X _i −X _i

, then |X _i

−EX _i

| ≤ 2i

and E(X _i

−EX _i

) ² ≤ EX _i

² ≤ CEX ² < ∞. From Lemma 3.1,

(34)

X ∞ i=1

a _ni (X _i

− EX _i

) → 0 a.s.

From (33), E ² XI(X ≥ i

) = O( i log i(log log i) ¹

), (i → ∞). And so (35)

X ∞ i=1

E ² XI(X ≥ i

) < ∞.

Obviously, (36)

X ∞ i=1

P (|X _i | ≥ i

) ≤ C X ∞ i=1

P (X ≥ i

)

and

(37) P (|X _i | ≥ i

, i.o.) = 0.

From(32), (35) and (37), [

X ∞ i=1

a _ni (X _i

− EX _i

)] ²

≤ 4(

X ∞ i=1

a ² _ni ) X ∞ i=1

(X _i ² I(|X _i | ≥ i

) +CE ² XI(|X _i | ≥ i

) → 0 a.s.

(38)

From (34) and (38), P _∞

i=1 a _ni X _i → 0 a.s..

Theorem 3.3. Suppose that {X _n , n ≥ 1} is a sequence of NOD random variables with EX _n = 0, X is a nonnegative random variable such that {X _n } ≺ X with EX < ∞ and {a _ni , i ≥ 1, n ≥ 1} is an array of constants satisfying

(39)

X ∞ i=1

|a _ni | = o(1) and sup

i≥1 i|a _ni | = o( 1 log n ).

Then

X ∞ i=1

a _ni X _i → 0 a.s..

Proof. Assume a _ni ≥ 0. Let X _i

= (−i) ∨ (X _i ∧ i) and X _i

= X _i − X _i

. From Lemma 3.1,

(40)

X ∞ i=1

a _ni (X _i

− EX _i

) → 0 a.s.

Since EX < ∞ and {X _n } ≺ X, we have (41) P {|X _i | ≥ i, i.o.} = 0.

From (39) and (41), we obtain

(42) |

X ∞ i=1

a _ni X _i

| ≤ X ∞

i=1

a _ni X ∞

i=1

|X _i |I(|X _i | ≥ i) → 0 a.s.

Again from (39), we get

| X ∞

i=1

a _ni EX _i

| ≤ (sup

i

E|X _i

|) X ∞ i=1

a _ni

≤ C(EX) X ∞

i=1

a _ni = o(1).

(43)

Finally, we obtain P _∞

i=1 a _ni X _i → 0 a.s. from (40), (42) and (43).

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Mi-Hwa Ko

Statistical Research Center for Complex Systems Seoul National University

Seoul 151-742, Korea

E-mail: [email protected]

Kwang-Hee Han

Department of Computer Science Howon University

Jeonbuk 573-718, Korea

E-mail: [email protected] Tae-Sung Kim

Department of Mathematics WonKwang University Jeonbuk 570-749, Korea

E-mail: [email protected]