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}=a X_n+Y_{n+1}, where a is a constant, stays non-negative for a long time. Assuming that the i.i.d. innovations Y_n take only two values +1 and -1 and 0 < a <= 2/3, we find the exact asymptotics of this probability and the weak limit of X_$ conditioned to stay non-negative. 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 E Y_n =0, where this distribution is uniform on the interval [0,3]. These properties are similar to those of the Bernoulli convolutions. To solve our problem, we employ a dynamical system defined by a certain linear mod 1 transform. Such mappings are well studied due to their use in expansions of numbers in non-integer bases, the so-called generalised beta-expansions. This is a joint work with V. Wachtel.