On a stochastic difference equation and a representation of non–negative infinitely divisible random variables

The present paper considers the stochastic difference equation Y n = A n Y n -1 + B n with i.i.d. random pairs ( A n , B n ) and obtains conditions under which Y n converges in distribution. This convergence is related to the existence of solutions of and ( A, B ) independent, and the convergence w.p. 1 of ∑ A 1 A 2 ··· A n -1 B n . A second subject is the series ∑ C n f ( T n ) with ( C n ) a sequence of i.i.d. random variables, ( T n ) the sequence of points of a Poisson process and f a Borel function on (0, ∞). The resulting random variable turns out to be infinitely divisible, and its Lévy–Hinčin representation is obtained. The two subjects coincide in case A n and C n are independent, B n = A n C n , A n = U 1/α n with U n a uniform random variable, f ( x ) = e − x /α .