A note on the Hajek–Renyi inequality for associated random variables

Statistics and Probability Letters 78 (2008) 885–889 www.elsevier.com/locate/stapro

A note on the Hajek–Renyi inequality for associated random variables Soo Hak Sung Department of Applied Mathematics, Pai Chai University, Taejon 302-735, South Korea Received 10 July 2006; received in revised form 3 January 2007; accepted 7 September 2007 Available online 20 November 2007

Abstract A Hajek–Renyi type inequality for associated random variables was obtained by Prakasa Rao [Prakasa Rao, B.L.S, 2002, Hajek–Renyi-type inequality for associated sequences. Statist. Probab. Lett. 57, 139–143]. In this paper, we improve the inequality. Some applications are also given. c 2007 Elsevier B.V. All rights reserved.

MSC: 60E15; 60F15

1. Introduction Let P {X n , n ≥ 1} be a sequence of random variables defined on a fixed probability space (Ω , F, P), and let n Sn = i=1 X i for n ≥ 1. The concept of (positively) associated random variables was introduced by Esary et al. (1967). A finite sequence {X i , 1 ≤ i ≤ n} is said to be associated if for any componentwise nondecreasing functions f and g on R n Cov( f (X 1 , . . . , X n ), g(X 1 , . . . , X n )) ≥ 0 whenever the covariance exists. An infinite sequence {X n , n ≥ 1} is said to be associated if every finite subfamily is associated. It is easy to see that if {X n } is a sequence of associated random variables, then the covariance is nonnegative. Let us recall that the independent random variables are associated and nondecreasing functions of associated random variables are also associated. Hajek and Renyi (1955) proved the following important inequality. If {X n , n ≥ 1} is a sequence of independent random variables with mean zero and finite second moments, and {bn , n ≥ 1} is a sequence of positive nondecreasing real numbers, then, for any  > 0, k   P ! X n m   j=1 j X E X 2j 1 X   −2 2 P  max + 2 EXj . >  ≤  2  m≤k≤n bk bm j=1 j=m+1 b j The inequality was studied by many authors (see Gan (1997), Liu et al. (1999) and Prakasa Rao (2002)). E-mail address: [email protected]. c 2007 Elsevier B.V. All rights reserved. 0167-7152/$ - see front matter doi:10.1016/j.spl.2007.09.015

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Recently, Prakasa Rao (2002) extended the Hajek–Renyi inequality to associated random variables. In this paper, we improve the inequality of Prakasa Rao. Using this result, we obtain the integrability of supremum and strong law of large numbers for associated random variables. 2. Hajek–Renyi inequality To prove the Hajek–Renyi inequality for associated random variables, the following lemma is needed. Matula (1996) proved the following lemma replacing the factor 2 by 8. Lemma P 2.1. Let X 1 , . . . , X n be associated random variables with mean zero and finite second moments, and let Si = ij=1 X j for 1 ≤ i ≤ n. Then, for any  > 0, P( max |Si | > ) ≤ 2 −2 E Sn2 . 1≤i≤n

Proof. Observe that max |Si |2 = max{| max Si |2 , | max (−Si )|2 }.

1≤i≤n

1≤i≤n

1≤i≤n

By Theorem 2 in Newman and Wright (1981) (see also Corollary 5 in Newman and Wright (1982)), we get E| max Si |2 ≤ E|Sn |2 . 1≤i≤n

Since −X 1 , . . . , −X n are associated random variables, we can replace Si by −Si in the above statement. That is, E| max (−Si )|2 ≤ E|Sn |2 . 1≤i≤n

Hence P( max |Si | > ) ≤ 1≤i≤n

1 1 2 E( max |Si |2 ) ≤ 2 {E| max Si |2 + E| max (−Si )|2 } ≤ 2 E Sn2 .  2 1≤i≤n 1≤i≤n 1≤i≤n   

Theorem 2.1. Let {X n , n ≥ 1} be a sequence of associated random variables with E X n = 0 and E X n2 < ∞, n ≥ 1. Let {bn , n ≥ 1} be a sequence of positive nondecreasing real numbers. Then, for any  > 0, ! ( ) k n n E X2 1 X X Cov(X j , S j−1 ) 8 X j P max X j >  ≤ 2 +2 . 2 1≤k≤n bk  b2j j=1 j=1 b j j=1 Proof. Without loss of generality, we may assume that bn ≥ 1 for all n ≥ 1. Let α =

√ 2. For i ≥ 0, define

Ai = {1 ≤ k ≤ n : α i ≤ bk < α i+1 }. Then Ai may be an empty set. When Ai 6= ∅, we let v(i) = max{k : k ∈ Ai }. Let tn be the index of the last nonempty set Ai . It is easy to see that α i ≤ bk ≤ bv(i) < α i+1 if k ∈ Ai . By Lemma 2.1, we have that ! ! k k 1 X 1 X P max X j >  = P X j >  max max 1≤k≤n bk 0≤i≤tn ,Ai 6=∅ k∈Ai bk j=1 j=1 ! tn k 1 X X ≤ P max X j >  k∈Ai bk j=1 i=0,Ai 6=∅ ! tn k X X 1 ≤ P max X j >  α i 1≤k≤v(i) j=1 i=0,Ai 6=∅  2  v(i) tn X X 2 1   X ≤ 2 E j j=1  α 2i i=0,Ai 6=∅

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=

=

2 2 2 2

v(i) 1 X {E X 2j + 2 Cov(X j , S j−1 )} 2i α j=1 i=0,A 6=∅ tn X i

n X

tn X

{E X 2j + 2 Cov(X j , S j−1 )}

j=1

i=0,Ai 6=∅,v(i)≥ j

1 , α 2i

(1)

where the maximum max0≤i≤tn ,Ai 6=∅ is taken for all 0 ≤ i ≤ tn such that Ai 6= ∅. Ptn 1 . Let i 0 = min{i : Ai 6= ∅, v(i) ≥ j}. Then b j ≤ bv(i0 ) < α i0 +1 by the Now we estimate i=0,A i 6=∅,v(i)≥ j α 2i definition of v(i). It follows that tn X i=0,Ai 6=∅,v(i)≥ j

∞ X 4 1 1 1 1 α2 1 = 2. < = < 1 1 2i 2i 2i 2 0 α α bj 1 − α2 α 1 − α2 b j i=i 0

Thus the result follows by (1) and (2).

(2)



Remark 2.1. Under the same conditions of Theorem 2.1, Prakasa Rao (2002) obtained the following result ) ! ( k n E X2 1 X X Cov(X j , X k ) 8 X j + . P max X j >  ≤ 2 2 1≤k≤n bk b j bk  j=1 j=1 b j 1≤ j6=k≤n Note that there was a typo in Prakasa Rao (2002) (the factor 4 should be 8). Since {bn , n ≥ 1} is a sequence of positive nondecreasing real numbers, the bound in Theorem 2.1 is dominated by ( ) n E X2 X Cov(X j , X k ) 8 X j . + b j bk  2 j=1 b2j 1≤ j6=k≤n Hence Theorem 2.1 improves the result of Prakasa Rao (2002). Using Theorem 2.1, we can obtain the following Hajek–Renyi type inequality for associated random variables. Theorem 2.2. Let {X n , n ≥ 1} be a sequence of associated random variables with E X n = 0 and E X n2 < ∞, n ≥ 1. Let {bn , n ≥ 1} be a sequence of positive nondecreasing real numbers. Then, for any  > 0 and for any positive integer m < n, ! m k n 1 X Cov(X j , S j ) 8 X 16 X X j >  ≤ 2 2 Cov(X j , S j ) + 2 . P max m≤k≤n bk  b  b2j m j=1 j=1 j=m+1 Proof. Observe that !     k 1 X |Sm |  |Sk − Sm |  + P max P max X j >  ≤ P max > > m≤k≤n bk m≤k≤n bk m≤k≤n 2 bk 2 j=1 =: I + I I. For I, we have by Markov’s inequality that   m |Sm |  4 8 X I ≤P > ≤ 2 2 E Sm2 ≤ 2 2 Cov(X j , S j ). bm 2  bm  bm j=1 For I I, we will apply Theorem 2.1 to {X m+i , 1 ≤ i ≤ n − m} and {bm+i , 1 ≤ i ≤ n − m}. Noting that P k X m+ j j=1 |Sk − Sm | max = max , m≤k≤n 1≤k≤n−m bk bm+k

(3)

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we have by Theorem 2.1 that ( ) 2 n−m X E X m+ X Cov(X m+ j , Sm+ j−1 − Sm ) 8 n−m j +2 II ≤ 2 2 2  bm+ j j=1 bm+ j j=1 ≤

X Cov(X m+ j , Sm+ j − Sm ) 16 n−m 2 2  j=1 bm+ j

=

n Cov(X j , S j − Sm ) 16 X  2 j=m+1 b2j

≤

n Cov(X j , S j ) 16 X ,  2 j=m+1 b2j

(4)

since Cov(X j , X k ) ≥ 0 by the definition of association. Hence the result follows by (3) and (4).  3. Applications In this section, we will obtain the integrability of supremum and strong law of large numbers for associated random variables. real numbers. Let {X n , n ≥ 1} be a Theorem 3.1. Let {bn , n ≥ 1} be a sequence of positive nondecreasing P 2 sequence of associated mean zero random variables satisfying ∞ j=1 Cov(X j , S j )/b j < ∞. If 0 < r < 2, then r E(supn≥1 (|Sn |/bn ) ) < ∞. Proof. By Theorem 2.1, we get ! r r ! Z ∞ Sn Sn P sup > t dt E sup = 0 n≥1 bn n≥1 bn ! ! r r Z 1 Z ∞ Sn Sn = P sup > t dt + P sup > t dt 0 1 n≥1 bn n≥1 bn Z ∞ X Cov(X j , S j ) ∞ 2 t − r dt < ∞. ≤ 1 + 16 2 b 1 j j=1 

So the result is proved.

Theorem 3.2. Let {bn , n ≥ 1} be a nondecreasing unbounded sequence of positive real numbers. Let {X n , n ≥ 1} be P 2 a sequence of associated mean zero random variables satisfying ∞ j=1 Cov(X j , S j )/b j < ∞. Then Sn /bn → 0 a.s. Proof. Observe that       Sn Sn ∞ Sn ∞ P ∪n=m >  = P ∪ N =m max >  = lim P max >  . N →∞ m≤n≤N bn m≤n≤N bn bn By Theorem 2.2, we have that   m N Sn Cov(X j , S j ) 8 X 16 X P max >  ≤ 2 2 . Cov(X j , S j ) + 2 m≤n≤N bn  bm j=1  j=m+1 b2j Hence we get by the Kronecker lemma that   ∞ Sn lim P ∪n=m >  = 0. m→∞ b n

So the result is proved.



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Corollary 3.1. Let {an , n ≥ 1} be a sequence of positive real numbers satisfying bn → ∞, where bn = n ≥ 1. Let {X n , n ≥ 1} be a sequence of associated mean zero random variables satisfying

Pn

i=1 ai ,

j ∞ X 1 X ai a j Cov(X i , X j ) < ∞. 2 j=1 b j i=1 Pn Then i=1 ai X i /bn → 0 a.s.

Proof. It is easy to see that {an X n , n ≥ 1} is a sequence of associated random variables. The result follows by Theorem 3.2.  Remark 3.1. Birkel (1989) proved Corollary 3.1 when an = 1, n ≥ 1. Matula (1996) proved Corollary 3.1 under an additional condition an /bn → 0. Prakasa Rao (2002) proved Theorems 3.1 and 3.2 under the stronger condition ∞ E X2 X j j=1

b2j

+

∞ X Cov(X j , X k ) < ∞. b j bk 1≤ j6=k

Acknowledgements The author would like to thank the referee for the helpful comments. This work was supported by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (MOST) (No. R01-2007-000-20053-0). References Birkel, T., 1989. A note on the strong law of large numbers for positively dependent random variables. Statist. Probab. Lett. 7, 17–20. Esary, J.D., Proschan, F., Walkup, D.W., 1967. Association of random variables, with applications. Ann. Math. Statist. 38, 1466–1474. Gan, S., 1997. The H`ajeck–R`enyi inequality for Banach space valued martingales and the p smoothness of Banach spaces. Statist. Probab. Lett. 32, 245–248. Hajek, J., Renyi, A., 1955. A generalization of an inequality of Kolmogorov. Acta Math. Acad. Sci. Hungar. 6, 281–284. Liu, J., Gan, S., Chen, P., 1999. The H`ajeck–R`enyi inequality for the NA random variables and its application. Statist. Probab. Lett. 43, 99–105. Matula, P., 1996. Convergence of weighted averages of associated random variables. Probab. Math. Statist. 16, 337–343. Newman, C.M., Wright, A.L., 1981. An invariance principle for certain dependent sequences. Ann. Probab. 9, 671–675. Newman, C.M., Wright, A.L., 1982. Associated random variables and martingale inequalities. Z. Wahrsch. Verw. Gebiete 59, 361–371. Prakasa Rao, B.L.S., 2002. Hajek–Renyi-type inequality for associated sequences. Statist. Probab. Lett. 57, 139–143.