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

Abstract

The present paper considers the stochastic difference equation Y_n = A_nY_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 ∑ A₁A₂ ··· A_n-1B_n. A second subject is the series ∑ C_nf(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_nC_n, A_n = U^1/α_n with U_n a uniform random variable, f(x) = e^−x/α.