The Asymptotic Behavior Of The Random Sums Of The Random Variables With The Extended Negatively Dependent Increment

Tail probability of random sums plays an important role in risk theory, queueing theory,renewal theory, etc. Let {X,Xk,k≥1} be sequences of r.v. with common distribution F(x),let Sn be n-th partial sum of {Xk,k≥1}, letτbe nonnegative integer-valued r.v., mutualindependent of {X,Xk,k≥1}, with the distribution Fτ(x). Zhang et al(2011) discussed theasymptotic behavior of the random sums of the r.v. with the NA increment. The extendednegative dependent(END) is more significant than the negative association(NA). Based onZhang's conclusion, we get the asymptotic behavior of the random sums of the r.v. withthe END increment. weakening the condition that the left tail of Xk is zero, prove that ashold. Besides wediscuss the situation Eτ=∞, then apply the result to estimating the finite ruin probabilitywith a constant rate, obtaining a uniform estimate when claims at a certain timeis END.