Survival probabilities of autoregressive processes

Technische Universität Braunschweig, Institut für Mathematische Stochastik, Pockelsstrasse 14, 38106 Braunschweig, Germany
baumgart@math.tu-berlin.de

Received: 31 May 2012
Revised: 8 October 2012

Abstract

Given an autoregressive process X of order p (i.e. X_n = a₁X_n−1 + ··· + a_pX_n−p + Y_n where the random variables Y₁, Y₂,... 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 a₁,..., a_p and the distribution of Y₁, 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.