Persistence of AR($1$) sequences with Rademacher innovations and linear mod $1$ transforms

Abstract

We study the probability that an AR(1) Markov chain $X_{n+1}=a{X}_n+{\xi }_{n+1}$, where $a\in (0,1)$ is a constant, stays non-negative for a long time. We find the exact asymptotics of this probability and the weak limit of $X_n$ conditioned to stay non-negative, assuming that the i.i.d.\ innovations ${\xi }_n$ take only two values $\pm 1$ and $a\le \frac{2}{3}$. This limiting distribution is quasi-stationary. It has no atoms and is singular with respect to the Lebesgue measure when $\frac{1}{2}<a\le \frac{2}{3}$, except for the case $a=\frac{2}{3}$ and $\mathbb{P}({\xi }_n=1)=\frac{1}{2}$, where this distribution is uniform on the interval $[0,3]$. This is similar to the properties of Bernoulli convolutions. For $0<a\le \frac{1}{2}$, the situation is much simpler, and the limiting distribution is a $\delta$-measure. To prove these results, we uncover a close connection between $X_n$ killed at exiting $[0,\infty )$ and the classical dynamical system defined by the piecewise linear mapping $x\mapsto \frac{1}{a}x+\frac{1}{2}(\mathrm{mod}1)$. Namely, the trajectory of this system started at $X_n$ deterministically recovers the values of the killed chain in reversed time. We use this fact to construct a suitable Banach space, where the transition operator of the killed chain has the compactness properties that allow us to apply a conventional argument of the Perron--Frobenius type.

50 pages