Dr. Vladislav Vysotsky (University of Sussex) – Persistence of AR(1) sequences with Rademacher innovations and linear mod 1 transforms
- Date
- @ MALL, 14:00
- Location
- MALL
- Speaker
- Dr. Vladislav Vysotsky
- Affiliation
- University of Sussex
- Category
- Probability
We study the probability that an AR(1) Markov chain $X_{n+1}=aX_n+\xi_{n+1}$, where $a$ is a constant, stays non-negative for a long time. Assuming that the i.i.d. innovations $\xi_n$ take only two values $\pm 1$ and $a \le \frac23$, we find the exact asymptotics of this probability and the weak limit of $X_n$ 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 $\frac12< a \le \frac23$, except for the case $a=\frac23$ and $P(\xi_n=1)=\frac12$, 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.