Slowly varying asymptotics for signed stochastic difference equations

Korshunov, Dmitry (2021) Slowly varying asymptotics for signed stochastic difference equations. In: A Lifetime of Excursions Through Random Walks and Lévy Processes : A Volume in Honour of Ron Doney's 80th Birthday. Progress in Probability . Birkhauser, Cham, pp. 245-257. ISBN 9783030833084

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Abstract

For a stochastic difference equation D n = A n D n−1 + B n which stabilises upon time we study tail distribution asymptotics for D n under the assumption that the distribution of log(1+|A1|+|B1|) is heavy-tailed, that is, all its positive exponential moments are infinite. The aim of the present paper is three-fold. Firstly, we identify the asymptotic behaviour not only of the stationary tail distribution but also of D n. Secondly, we solve the problem in the general setting when A takes both positive and negative values. Thirdly, we get rid of auxiliary conditions like finiteness of higher moments introduced in the literature before.