Persistence exponents via perturbation theory: AR(1)-processes

Abstract

For AR(1)-processes $X_n=\rho X_{n-1}+{\xi }_n$, $n\in \mathbb{N}$, where $\rho \in \mathbb{R}$ and $({\xi }_i{)}_{i\in \mathbb{N}}$ is an i.i.d. sequence of random variables, we study the persistence probabilities $\mathbb{P}(X_0\ge 0,\dots ,X_N\ge 0)$ for $N\to \infty$. For a wide class of Markov processes a recent result [Aurzada, Mukherjee, Zeitouni; arXiv:1703.06447; 2017] shows that these probabilities decrease exponentially fast and that the rate of decay can be identified as an eigenvalue of some integral operator. We discuss a perturbation technique to determine a series expansion of the eigenvalue in the parameter $\rho$ for normally distributed AR(1)-processes.

Version 2 contains an appendix that develops the relevant concepts from perturbation theory for linear operators

References in corpus (3)

• General criteria for the study of quasi-stationarity
• Survival probabilities of weighted random walks
• Persistence versus stability for auto-regressive processes

Cited by in corpus (1)

• Persistence versus stability for auto-regressive processes