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Paper Information

Journal:   JOURNAL OF SCIENCES ISLAMIC REPUBLIC OF IRAN   FALL 2008 , Volume 19 , Number 4; Page(s) 357 To 363.
 
Paper: 

ASYMPTOTIC BEHAVIOR OF WEIGHTED SUMS OF WEAKLY NEGATIVE DEPENDENT RANDOM VARIABLES

 
 
Author(s):  RANJBAR V., AMINI Y.M.*, BOZORGNIA A.
 
* DEPARTMENT OF STATISTICS, FACULTY OF MATHEMATICAL SCIENCES, FERDOWSI UNIVERSITY OF MASHHAD, MASHHAD, ISLAMIC REPUBLIC OF IRAN
 
Abstract: 
Let {Xj ; j³1} be a sequence of weakly negative dependent (denoted by, WND) random variables with common distribution function F and let {qj ; j³1} be other sequence of positive random variables independent of {Xj ; j³1} and P[a£qj £b]=1 for some 0<a£b<¥ and for all j³1. In this paper, we study the asymptotic behavior of the tail probabilities of the maximum, weighted sums, randomly weighted sums and randomly indexed weighted sums of heavy-tailed weakly negative dependent random variables, say, max1£j£nXj, Snj=1 cjXj , Snj=1 qjXj , and SNj=1 qjXj , respectively, where {cj ; 1£j£n} are n bounded positive real numbers and N is a nonnegative integer-valued random variables, independent of qi and Xi for all i³1. In fact, for a large class of heavy-tailed distribution functions, we show that the asymptotic relations,
P[max1
£j£n qjXj>x]~P[Snj=1 qjXj>x]~Snj=1 P[qjXj>x],
hold as x
®¥. Finally, if E(N)<¥ and also {qj ; j³1} is a sequence of identical independent positive random variables, then we prove that P[SNj=1 qjXj>x]~E(N). P[q1X1>x], as x®¥.
 
Keyword(s): WEAKLY NEGATIVE DEPENDENT, HEAVY-TAILED, ASYMPTOTIC BEHAVIOR, WEIGHTED SUMS
 
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