Recurrence and transience of contractive autoregressive processes and related Markov chains

TL;DR

This paper provides criteria for recurrence and transience in various stochastic processes, including autoregressive and branching processes, based on Lyapunov exponents and distribution functions.

Contribution

It offers a unified characterization of recurrence and transience for multiple classes of stochastic processes using Lyapunov exponents and distribution criteria.

Findings

Criteria for recurrence and transience are established.

The maximal Lyapunov exponent determines process behavior.

Results apply to autoregressive, branching, and max-autoregressive processes.

Abstract

We characterize recurrence and transience of nonnegative multivariate autoregressive processes of order one with random contractive coefficient matrix, of subcritical multitype Galton-Watson branching processes in random environment with immigration, and of the related max-autoregressive processes and general random exchange processes. Our criterion is given in terms of the maximal Lyapunov exponent of the coefficient matrix and the cumulative distribution function of the innovation/immigration component.