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 α-stable innovations. We write Λ(a,α) for the symmetric exponential persistence rate and Λ(a,α,ρ) for the rate of a strictly stable law with positivity parameter ρ. The entire chain admits an exact representation through a single stable Lévy process observed on a geometrically expanding time grid. In the symmetric case, comparison with continuous half-line survival gives Λ(a,α) ≤ α/2log(1/a). For 0 < α< 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 Λ(a,α) asymp log(1/a) as a ↑ 1 and shows that Λ(a,α)/log(1/a) converges to a limit in (0,α/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 α=2; for 0 < α< 2, rescued crossings between observations remain the obstacle. For asymmetric strictly stable innovations with α≠ 1 and admissible ρ∈(0,1), the corresponding bounds are Λ(a,α,ρ) ≤ α(1−ρ)log(1/a) and Λ(a,α,ρ) asymp log(1/a). At α=2, only the symmetric case ρ=1/2 occurs.
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BibTeX
@misc{mendonca2026nearunitroot,
title = {{Near-unit-root persistence of symmetric and asymmetric stable autoregressive sequences}},
author = {José Ricardo G. Mendonça and Boubaker Smii},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.17927},
eprint = {2608.17927},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.17927},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Near-unit-root persistence of symmetric and asymmetric stable autoregressive sequences
AU - José Ricardo G. Mendonça
AU - Boubaker Smii
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2608.17927
UR - https://arxiv.org/abs/2608.17927
AB - 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 α-stable innovations. We write Λ(a,α) for the symmetric exponential persistence rate and Λ(a,α,ρ) for the rate of a strictly stable law with positivity parameter ρ. The entire chain admits an exact representation through a single stable Lévy process observed on a geometrically expanding time grid. In the symmetric case, comparison with continuous half-line survival gives Λ(a,α) ≤ α/2log(1/a). For 0 < α< 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 Λ(a,α) asymp log(1/a) as a ↑ 1 and shows that Λ(a,α)/log(1/a) converges to a limit in (0,α/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 α=2; for 0 < α< 2, rescued crossings between observations remain the obstacle. For asymmetric strictly stable innovations with α≠ 1 and admissible ρ∈(0,1), the corresponding bounds are Λ(a,α,ρ) ≤ α(1−ρ)log(1/a) and Λ(a,α,ρ) asymp log(1/a). At α=2, only the symmetric case ρ=1/2 occurs.
ER -