Near-unit-root persistence of symmetric and asymmetric stable autoregressive sequences

Abstract

For an AR($1$) sequence with coefficient $0<a<1$, the probability of staying above zero decays exponentially in the number of steps; at the unit root $a=1$ it decays polynomially, with order $n^{-1/2}$ for symmetric increments. We study this transition for AR($1$) sequences driven by symmetric and asymmetric $\alpha$-stable innovations. We write $\Lambda(a,\alpha)$ for the symmetric exponential persistence rate and $\Lambda(a,\alpha,\rho)$ for the rate of a strictly stable law with positivity parameter $\rho$. The entire chain admits an exact representation through a single stable L\'{e}vy process observed on a geometrically expanding time grid. In the symmetric case, 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 concerning regularly varying innovation tails. 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 $\Lambda(a,\alpha)/\log{(1/a)}$ converges to a limit in $(0,\alpha/2]$ equal to the supremum of that ratio over $0<a<1$. A Lamperti transformation reduces identification of this constant to a dense-sampling problem for a stationary stable Ornstein-Uhlenbeck process. Existing Gaussian theory determines the sharp value at $\alpha=2$; for $0<\alpha<2$, rescued crossings between observations remain the obstacle. For asymmetric strictly stable innovations with $\alpha \neq 1$ and admissible $\rho\in(0,1)$, the corresponding bounds are $\Lambda(a,\alpha,\rho) \leq \alpha(1-\rho)\log{(1/a)}$ and $\Lambda(a,\alpha,\rho) \asymp \log{(1/a)}$. At $\alpha=2$, only the symmetric case $\rho=1/2$ occurs.