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

We study the probability that an AR(1) Markov chain X_n+1=aX_n+ξ_n+1, where a∈(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 ξ_n take only two values ±1 and a ≤2/3. This limiting distribution is quasi-stationary. It has no atoms and is singular with respect to the Lebesgue measure when 1/2< a ≤2/3, except for the case a=2/3 and 𝐏(ξ_n=1)=1/2, where this distribution is uniform on the interval [0,3]. This is similar to the properties of Bernoulli convolutions. It turns out that for the ±1 innovations there is a close connection between X_n killed at exiting [0, ∞) and the classical dynamical system defined by the piecewise linear mapping x ↦1/a x + 1/2 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 property to construct a suitable Banach space, where the transition operator of the killed chain has the compactness properties that let us apply a conventional argument of Perron–Frobenius type. The difficulty in finding such space stems from discreteness of the innovations.