Near-unit-root persistence of symmetric stable autoregressive sequences

Abstract

Persistence changes character as an autoregressive coefficient approaches one: for each fixed $0 < a < 1$, survival above zero decays exponentially, whereas at the unit root symmetric random-walk survival is of order $n^{-1/2}$. We study this transition for AR($1$) sequences driven by symmetric $\alpha$-stable innovations and write $\Lambda(a,\alpha)$ for their exponential persistence rate. The entire chain admits an exact representation through a single stable L\'{e}vy process observed on a geometrically expanding time grid. Comparison with continuous half-line survival gives $\Lambda(a,\alpha) \leq \frac{\alpha}{2}\log{(1/a)}$. For $0 < \alpha < 2$, this bound disproves the stable specialization of a conjecture of Hinrichs, Kolb and Wachtel for regularly varying innovation tails. To obtain a lower bound of the same near-unit order, we combine stable closure under subsampling with a monotonicity coupling. This proves $\Lambda(a,\alpha) \asymp \log{(1/a)}$ as $a \uparrow 1$ and shows that the ratio $\Lambda(a,\alpha)/\log{(1/a)}$ converges to a limit in $(0,\alpha/2]$, equal to its supremum over $0 < a < 1$. Finally, a Lamperti transformation reduces identification of this constant to a dense-sampling persistence problem for a stationary stable Ornstein--Uhlenbeck process. Existing Gaussian theory determines the sharp value at $\alpha=2$. For $0 < \alpha < 2$, identifying the value requires controlling paths that cross below zero and return above zero between consecutive observations.