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 Lévy process observed on a geometrically expanding time grid. Comparison with continuous half-line survival gives $\Lambda (a,\alpha )\le \frac{\alpha }{2}\log (1/a)$. For $0<\alpha <2$, this bound disproves the stable specialization of a conjecture of Hinrichs, Kolb and Wachtel for regularly varying innovation tails. \rev{Combining stable closure under subsampling with a monotonicity coupling yields a lower bound of the same near-unit order.} 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.

AMSart style, 16 pages, 20 references. Version v2 improves the proofs of Lemmas 5.1 and 5.7 and provides additional clarification in Remark 3.6 regarding the scope of the counterexample to the HKW conjecture