Randomly Weighted Sums of Subexponential Random Variables with Application to Ruin Theory |
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| Authors: | Qihe Tang Gurami Tsitsiashvili |
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| Institution: | (1) Department of Quantitative Economics, University of Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands;(2) Institute of Applied Mathematics, Far Eastern Scientific Center, Russian Academy of Sciences, 690068 Vladivostok, Russia |
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| Abstract: | Let {X
k
, 1  k n} be n independent and real-valued random variables with common subexponential distribution function, and let { k, 1 k n} be other n random variables independent of {X
k
, 1 k n} and satisfying a
k
b for some 0 < a b <  for all 1 k n. This paper proves that the asymptotic relations P (max1 m n 
k=1
m
k
X
k
> x)  P (sum
k=1
n
k
X
k
> x) sum
k=1
n
P (
k
X
k
> x) hold as x  . In doing so, no any assumption is made on the dependence structure of the sequence {
k
, 1 k n}. An application to ruin theory is proposed. |
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| Keywords: | asymptotics dominated variation ruin probability subexponentiality uniformity |
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