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

Vladislav Vysotsky, Vitali Wachtel

Abstract

We study the probability that an AR(1) Markov chain $X_{n+1}=aX_n+ξ_{n+1}$, 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 $ξ_n$ take only two values $\pm1$ and $a \le \frac23$. 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 $\mathbb{P}(ξ_n=1)=\frac12$, where this distribution is uniform on the interval $[0,3]$. This is similar to the properties of Bernoulli convolutions. For $0 < a \le \frac12$, the situation is much simpler, and the limiting distribution is a $δ$-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 $x \mapsto \frac1a x + \frac12 \pmod 1$. 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.

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

Abstract

We study the probability that an AR(1) Markov chain , where is a constant, stays non-negative for a long time. We find the exact asymptotics of this probability and the weak limit of conditioned to stay non-negative, assuming that the i.i.d.\ innovations take only two values and . This limiting distribution is quasi-stationary. It has no atoms and is singular with respect to the Lebesgue measure when , except for the case and , where this distribution is uniform on the interval . This is similar to the properties of Bernoulli convolutions. For , the situation is much simpler, and the limiting distribution is a -measure. To prove these results, we uncover a close connection between killed at exiting and the classical dynamical system defined by the piecewise linear mapping . Namely, the trajectory of this system started at 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.
Paper Structure (18 sections, 14 theorems, 247 equations)

This paper contains 18 sections, 14 theorems, 247 equations.

Table of Contents

  1. Introduction and main results
  2. Motivation
  3. Main results
  4. Outline of the approach and generalizations
  5. Deterministic dynamics under time reversal and three approaches to persistence
  6. Deterministic behaviour in reverse time
  7. Reduction to a finite chain when $\varkappa'_a<\infty$
  8. Compactness approach
  9. Transfer operator approach
  10. Existence of a positive solution to equation \ref{['eq: lambda']}
  11. Spectral properties of the killed transition operator on $U$
  12. Properties of the mapping $a \mapsto \lambda_a$
  13. Convergence to the quasi-stationary distribution $\nu_a$ and its properties
  14. Convergence of the conditional distributions
  15. Support and non-atomicity of $\nu_a$
  16. ...and 3 more sections

Key Result

Theorem 1

Let $\{X_n\}$ be a Markov chain defined by equation AR-def with some $a \in (\frac{1}{2}, \frac{2}{3}]$. Assume that the innovations $\{\xi_n\}$ satisfy Rad-assump with some $p \in (0,1)$. Then there exists a constant $c \in (0,1)$ such that, uniformly in $x \in [0,\frac{1}{1-a}]$, we have as $n \to \infty$, where $\lambda_a=\lambda_a(p) > p$ is the unique positive solution to and with $\delta_

Theorems & Definitions (34)

  • Theorem 1
  • Corollary 2
  • Remark 3
  • Proposition 4
  • Remark 5
  • Proposition 6
  • proof
  • Corollary 7
  • proof
  • Remark 8
  • ...and 24 more