ON PERPETUITIES WITH LIGHT TAILS

In this paper we consider the asymptotics of logarithmic tails of a perpetuity R (sic) Sigma(j)=1(infinity) Qj Pi(j-1)(k=1) M-k, where (M-n, Q(n))(n=1)(infinity) are independent and identically distributed copies of (M, Q), for the case when P(M is an element of [0, 1)) = 1 and Q has all exponential moments. If M and Q are independent, under regular variation assumptions, we find the precise asymptotics of - log P(R > x) as x -> infinity. Moreover, we deal with the case of dependent M and Q, and give asymptotic bounds for - log P(R > x). It turns out that the dependence structure between M and Q has a significant impact on the asymptotic rate of logarithmic tails of R. Such a phenomenon is not observed in the case of heavy-tailed perpetuities.