PERPETUITIES WITH THIN TAILS

We investigate the behaviour of P(R greater than or equal to r) and P( R less than or equal to-r) as r-->infinity for the random variable R:= Sigma(n=1)(infinity) Q(n) Pi(k=1)(n=1) M(k), where ((Q(k), M(k)))(k is an element of N) is an independent, identically distributed sequence with P(-1 less than or equal to M less than or equal to 1)=1. Random v ariables of this type appear in insurance mathematics, as solutions of stochastic difference equations, in the analysis of probabilistic alg orithms and elsewhere. Exponential and Poissonian tail behaviour can a rise.