Near-unit-root persistence of symmetric stable autoregressive sequences

Technical Summary: Near-Unit-Root Persistence of Symmetric Stable Autoregressive Sequences

Problem Statement
The paper investigates the persistence probability of a symmetric $\alpha$-stable autoregressive process of order 1 (AR(1)), defined by the recursion $X_n = aX_{n-1} + \xi_n$ with $0 < a < 1$, where $\xi_n$ are i.i.d. symmetric $\alpha$-stable innovations ($0 < \alpha \le 2$). The central object of study is the persistence probability $Q_n(x; a, \alpha, \sigma) = P_x(X_1 > 0, \dots, X_n > 0)$ and its exponential decay rate $\Lambda(a, \alpha)$.

The specific focus is the "near-unit-root" regime where the autoregressive coefficient $a$ approaches 1. In this limit, the discrete chain approximates a continuous-time stable Ornstein–Uhlenbeck (OU) process. The authors aim to characterize the asymptotic behavior of $\Lambda(a, \alpha)$ as $a \uparrow 1$, specifically determining the leading order term and the precise constant governing the transition from exponential decay (for fixed $a < 1$) to the polynomial decay ($n^{-1/2}$) observed at the unit root ($a=1$).

Methodology
The authors employ a combination of probabilistic representations, coupling arguments, and subsampling techniques:

1. Geometric-Time Representation: The AR(1) sequence is represented exactly as a symmetric $\alpha$-stable Lévy process $L_t$ sampled on a geometrically expanding time grid $\{u_n\}$. Specifically, $(X_n)_{n \ge 0} \stackrel{d}{=} (a^n(x + L_{u_n}))_{n \ge 0}$, where $u_n$ grows exponentially. This transforms the discrete persistence problem into a question about the survival of a stable path on a specific grid.
2. Lamperti Transformation: The problem is mapped to a stationary symmetric stable Ornstein–Uhlenbeck process $R_t = e^{-t/\alpha}L_{e^t}$ sampled at a uniform mesh $h$. This establishes a bijection between the AR(1) coefficient $a$ and the sampling mesh $h = \alpha \log(1/a)$.
3. Subsampling and Monotonicity: The authors utilize the closure of stable laws under subsampling. By observing the process every $m$-th step, they derive a new AR(1) sequence with coefficient $a^m$. Combined with a monotonicity coupling (showing that survival probability increases with $a$), this yields lower bounds for the persistence rate.
4. Continuous-Time Comparison: Upper bounds are derived by comparing the discrete grid survival to the continuous survival of the underlying stable Lévy process or the stationary OU process.

Key Contributions and Results

• Disproof of a Conjecture: The paper disproves a conjecture by Hinrichs, Kolb, and Wachtel regarding the persistence rate for innovations with regularly varying tails. While their conjecture suggested $\Lambda(a, \alpha) \sim \alpha \log(1/a)$ for symmetric $\alpha$-stable innovations (where the tail index is $\alpha$), the authors prove that for $0 < \alpha < 2$, the rate is strictly smaller:
  $$\Lambda(a, \alpha) \le \frac{\alpha}{2} \log(1/a).$$
  This discrepancy arises because the "big-jump" mechanism (dominant in high-threshold first passage) is subdominant to the continuous-survival mechanism (dominated by the Sparre–Andersen exponent $1/2$) in the near-unit-root regime.

• Asymptotic Characterization: The authors establish that as $a \uparrow 1$:
  $$\Lambda(a, \alpha) \asymp \log(1/a).$$
  More precisely, the ratio $G_\alpha(a) = \Lambda(a, \alpha) / \log(1/a)$ converges to a limit $G^*_\alpha$ as $a \uparrow 1$. This limit satisfies $G^*_\alpha \in (0, \alpha/2]$.

• Reduction to a Dense-Sampling Problem: Through the Lamperti transform, the identification of the constant $G^*_\alpha$ is reduced to a dense-sampling persistence problem for a stationary stable OU process. Specifically, $G^*_\alpha = \alpha \lim_{h \downarrow 0} \frac{\lambda_\alpha(h)}{h}$, where $\lambda_\alpha(h)$ is the persistence exponent of the OU process sampled at mesh $h$.

• Gaussian Case ($\alpha=2$): For the Gaussian case, existing theory (Feldheim et al.) confirms that the limit is sharp, i.e., $G^*_2 = 1$ (implying $\Lambda(a, 2) \sim \log(1/a)$). This recovers the continuous-time persistence rate of the OU process.

• Open Problem for $0 < \alpha < 2$: The paper leaves the exact value of $G^*_\alpha$ for $0 < \alpha < 2$ as an open conjecture. The authors conjecture that $G^*_\alpha = \alpha/2$, which would imply that the discrete sampling rate converges to the continuous-time rate even for jump-driven processes. However, they explicitly state that proving this requires controlling "rescued" sub-mesh crossings (paths that dip below zero and return between observations), a technical challenge not yet resolved.

Significance and Scope
The paper provides a rigorous framework for understanding the persistence of stable AR(1) processes near the unit root, correcting previous assumptions about the influence of heavy tails on the decay rate. By establishing the upper bound $\alpha/2 \log(1/a)$, the authors demonstrate that the persistence behavior is governed by the continuous path properties rather than the tail index alone.

The work is modest regarding the final value of the constant for non-Gaussian cases. It does not claim to have solved the dense-sampling limit for $0 < \alpha < 2$ but rather identifies the precise mathematical obstacle: the need to control the joint distribution of overshoots and return times in the stable Lévy process. The significance lies in the exact geometric-time representation, the disproof of the regular-variation conjecture for this specific regime, and the reduction of the problem to a well-defined, albeit difficult, dense-sampling question for stable OU processes.