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 conditions
where 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.
Acknowledgements
We thank the reviewer for their constructive comments, which helped us to improve the manuscript.
Notes
No potential conflict of interest was reported by the authors.
1 That is , where
2 We will also use the abbreviation: Lipschitz iterated system
3 The convergence follows from the subadditive ergodic theorem.