Persistence versus stability for auto-regressive processes

Abstract

The stability of an Auto-Regressive (AR) time sequence of finite order $L$, is determined by the maximal modulus $r^\star$ among all zeros of its generating polynomial. If $r^\star<1$ then the effect of input and initial conditions decays rapidly in time, whereas for $r^\star>1$ it is exponentially magnified (with constant or polynomially growing oscillations when $r^\star=1$). Persistence of such AR sequence (namely staying non-negative throughout $[0,N]$) with decent probability, requires the largest positive zero of the generating polynomial to have the largest multiplicity among all zeros of modulus $r^\star$. These objects are behind the rich spectrum of persistence probability decay for AR$_L$ with zero initial conditions and i.i.d. Gaussian input, all the way from bounded below to exponential decay in $N$, with intermediate regimes of polynomial and stretched exponential decay. In particular, for AR$_3$ the persistence decay power is expressed via the tail probability for Brownian motion to stay in a cone, exhibiting the discontinuity of such power decay between the AR$_3$ whose generating polynomial has complex zeros of rational versus irrational angles.