Stochastic recursions: between Kesten's and Grincevičius-Grey's assumptions

Abstract

We study the stochastic recursion $X_n={\Psi }_n(X_{n-1})$, where $({\Psi }_n{)}_{n\ge 1}$ is a sequence of i.i.d. random Lipschitz mappings close to the random affine transformation $x\mapsto Ax+B$. We describe the tail behaviour of the stationary solution $X$ under the assumption that there exists $\alpha >0$ such that $\mathbb{E}|A{|}^{\alpha }=1$ and the tail of $B$ is regularly varying with index $-\alpha <0$. We also find the second order asymptotics of the tail of $X$ when $\Psi (x)=Ax+B$.

31 pages. Presentation of results and the whole manuscript have been reworked substantially. Part of the previous version of manuscript was moved to arXiv:1812.04496 (Electron. Commun. Probab. 23 (2018))

References in corpus (4)

• On perpetuities with gamma-like tails
• On perpetuities with light tails
• The left tail of renewal measure
• A renewal theorem and supremum of a perturbed random walk

Cited by in corpus (1)

• On perpetuities with light tails