On the Kesten–Goldie constant

Abstract

We consider the stochastic difference equation on where is an i.i.d. sequence of random variables and is an initial distribution. Under mild contractivity hypotheses the sequence converges in law to a random variable S, which is the unique solution of the random difference equation . We prove that under the Kesten–Goldie conditionswhere is the Kesten–Goldie constant is the Cramér coefficient of and . Thus, on one side we describe the behaviour of the th moments of the process , and on the other we obtain an alternative formula for . The results are further extended to a class of Lipschitz iterated systems and to a multidimensional setting.