Survival probabilities of autoregressive processes

TL;DR

This paper investigates the long-term survival probabilities of autoregressive processes, analyzing how the probability of staying below a barrier decays over time based on process parameters and noise distribution.

Contribution

It provides new conditions characterizing the asymptotic decay of survival probabilities for AR(p) processes, with detailed focus on AR(2) models.

Findings

Survival probability can decay polynomially, faster, or stabilize depending on coefficients and noise.

Conditions are established for different decay regimes of the survival probability.

Special analysis is provided for AR(2) processes.

Abstract

Given an autoregressive process X of order p (i.e. X_n = a_1 X_{n-1} + ...+ a_p X_{n_p} + Y_n where the random variables Y_1, Y_2, ... are i.i.d.), we study the asymptotic behaviour of the probability that the process does not exceed a constant barrier up to time N (survival or persistence probability). Depending on the coefficients a_1,...,a_p and the distribution of Y_1, we state conditions under which the survival probability decays polynomially, faster than polynomially or converges to a positive constant. Special emphasis is put on AR(2) processes.