Slowly varying asymptotics for signed stochastic difference equations

For a stochastic difference equation \(D_n=A_nD_{n-1}+B_n\) which stabilises upon time we study tail distribution asymptotics of \(D_n\) under the assumption that the distribution of \(\log(1+|A_1|+|B_1|)\) is heavy-tailed, that is, all its positive exponential moments are infinite. The aim of the p...

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Bibliographic Details
Published inarXiv.org
Main Author Korshunov, Dmitry
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 27.07.2020
Subjects
Asymptotic properties
Difference equations
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Summary:For a stochastic difference equation \(D_n=A_nD_{n-1}+B_n\) which stabilises upon time we study tail distribution asymptotics of \(D_n\) under the assumption that the distribution of \(\log(1+|A_1|+|B_1|)\) 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 used in the literature before.
ISSN:2331-8422