Persistence of AR( 1 $1$ 1) sequences with Rademacher innovations and linear mod 1 $1$ 1 transforms
Abstract
Abstract We study the probability that an AR(1) Markov chain X n + 1 = a X n + ξ n + 1 $X_{n+1}=aX_n+\xi _{n+1}$ upper X Subscript n plus 1 Baseline equals a upper X Subscript n Baseline plus xi Subscript n plus 1 , where a ∈ ( 0 , 1 ) $a\in (0,1)$ a element of left parenthesis 0 comma 1 right parenthesis 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 $X_n$ upper X Subscript n conditioned to stay non-negative, assuming that the independent and identically distributed innovations ξ n $\xi _n$ xi Subscript n take only two values ± 1 $\pm 1$ plus or minus 1 and a ≤ 2 3 $a \le \tfrac 23$ a less than or equals two thirds . 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 $\t
Fields of Study & Subjects
Mathematical Dynamics and Fractals
Stochastic processes and statistical mechanics
Markov Chains and Monte Carlo Methods
Publication details
- Journal / source
- Ergodic Theory and Dynamical Systems
- Publisher
- Cambridge University Press (CUP)
- Volume
- 46
- Issue
- 9
- Pages
- 2361–2412
- Publication date
- 7 April 2026
- Language
- EN