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 $n<sup>{-1/2}$. We study this transition for AR($1$) sequences driven by symmetric $α$-stable innovations and write $Λ(a,α)$ 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 $Λ(a,α) \leq \fracα{2}\log{(1/a)}$. For $0 < α< 2$, 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 $Λ(a,α) \asymp \log{(1/a)}$ as $a \uparrow 1$ and shows that the ratio $Λ(a,α)/\log{(1/a)}$ converges to a limit in $(0,α/2]$, 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 $α=2$. For $0 < α< 2$, identifying the value requires controlling paths that cross below zero and return above zero between consecutive observations.

• The paper proves that the persistence rate (a,) is asymptotically proportional to log(1/a) as a 1, with 0 < lim /log(1/a) /2.
• It represents the stable AR(1) chain as a stable Lévy process observed on a geometrically expanding time grid, yielding the upper bound (a,) /2 log(1/a) and showing that the result is independent of the starting value and innovation scale.
• The analysis disproves the stable-law version of the HKW tail conjecture, identifies the sharp constant with a dense-sampling persistence problem for a stable Ornstein–Uhlenbeck process, and settles it only for the Gaussian case =2, where the constant equals 1.

Setting and main object

The paper studies the persistence probability of the AR(1) recursion $X_n = aX_{n-1} + \xi_n$ with $0 < a < 1$, started at $x > 0$, where the innovations $(\xi_n)$ are i.i.d. continuous symmetric $\alpha$-stable variables with characteristic function $\exp(-\sigma^\alpha|q|^\alpha)$, $0 < \alpha \le 2$. The quantity of interest is

$$Q_n(x;a,\alpha,\sigma) = P_x(X_1 > 0, \ldots, X_n > 0),$$

and its exponential rate $$\Lambda(a,\alpha) = -\lim_{n\to\infty} n^{-1}\log Q_n$$, whose existence for all starting points follows from Hinrichs–Kolb–Wachtel (2608.17927). The rate is independent of both $x > 0$ and the scale $0 < a < 1$0. The regime of interest is the near-unit-root limit $0 < a < 1$1: at $0 < a < 1$2 the chain is a symmetric random walk whose survival decays polynomially (Sparre Andersen: $0 < a < 1$3), while for each fixed $0 < a < 1$4 survival is exponential. The paper determines the order of $0 < a < 1$5 as $0 < a < 1$6 and reduces the sharp constant to a dense-sampling problem.

A structural fact underlies everything: the chain is the exact skeleton of a stable Ornstein–Uhlenbeck process sampled on a geometrically expanding time grid. Writing $0 < a < 1$7, the authors prove the path-space identity

$0 < a < 1$8

with $0 < a < 1$9 a standard symmetric $x > 0$0-stable Lévy process. Since $x > 0$1, persistence of the chain equals positivity of a single stable path at the deterministic times $x > 0$2. The grid is asymptotically uniform in logarithmic time with spacing $x > 0$3, so $x > 0$4 is simultaneously a dense-sampling limit in log-time.

Upper bounds and a counterexample

Two complementary upper bounds are established. First, continuous survival of $x > 0$5 up to time $x > 0$6 implies grid survival, and the classical small-deviation estimate for suprema of stable processes (Bingham; Doney–Savov) gives $x > 0$7 as $x > 0$8, uniformly bounded by $x > 0$9. Combining this with the explicit form of $(\xi_n)$0 yields

$(\xi_n)$1

together with an amplitude lower bound $(\xi_n)$2. Second, an association argument (the events $(\xi_n)$3 are increasing in the innovations, which have nonnegative coefficients in $(\xi_n)$4) gives $(\xi_n)$5, hence $(\xi_n)$6.

The $(\xi_n)$7 bound has a notable consequence: it disproves the stable specialization of a conjecture of Hinrichs, Kolb and Wachtel. That conjecture predicted $(\xi_n)$8 whenever the innovation right tail is regularly varying with index $(\xi_n)$9; for symmetric stable laws $\alpha$0, but the proven bound gives only $\alpha$1, strictly smaller for every $\alpha$2. The paper notes that the conjecture's proved version requires additional decay of the slowly varying factor, which stable laws do not satisfy; heuristically, the value $\alpha$3 reflects a big-jump mechanism that is dominated here by the continuous-survival contribution carrying the universal Sparre Andersen exponent $\alpha$4. The same comparison yields the continuous-time result that the stable Ornstein–Uhlenbeck process $\alpha$5 has persistence rate exactly $\alpha$6.

Lower bound via subsampling and monotonicity

The matching lower bound rests on two ingredients. First, subsampling every $\alpha$7-th observation produces another symmetric stable AR(1) chain — stability closes the block sums $\alpha$8 within the family, up to scale — with coefficient $\alpha$9. Dropping the intervening positivity constraints gives the inequality $\exp(-\sigma^\alpha|q|^\alpha)$0, equivalently $\exp(-\sigma^\alpha|q|^\alpha)$1. Second, a coupling shows $\exp(-\sigma^\alpha|q|^\alpha)$2 is nondecreasing, hence $\exp(-\sigma^\alpha|q|^\alpha)$3 is nonincreasing. The remark on where stability enters is precise: monotonicity and the constraint-dropping inclusion hold for arbitrary innovations; stability is used only to identify the subsampled chain within the same family.

Combining these, for any fixed $\exp(-\sigma^\alpha|q|^\alpha)$4 and $\exp(-\sigma^\alpha|q|^\alpha)$5, choosing $\exp(-\sigma^\alpha|q|^\alpha)$6 gives

$\exp(-\sigma^\alpha|q|^\alpha)$7

Consequently $\exp(-\sigma^\alpha|q|^\alpha)$8 as $\exp(-\sigma^\alpha|q|^\alpha)$9, and the ratio $0 < \alpha \le 2$0 satisfies

$0 < \alpha \le 2$1

The existence of this limit and its identification with the global supremum follow directly; the value of $0 < \alpha \le 2$2 is left open for $0 < \alpha \le 2$3.

Lamperti reduction and the Gaussian benchmark

Applying the Lamperti transform $0 < \alpha \le 2$4 yields a stationary Markov process with marginal law $0 < \alpha \le 2$5 and transitions $0 < \alpha \le 2$6. Its discretization at mesh $0 < \alpha \le 2$7 is precisely the stationary stable AR(1) chain with coefficient $0 < \alpha \le 2$8, and the stationary-start rate of Proposition (ii) of HKW identifies

$0 < \alpha \le 2$9

where $$Q_n(x;a,\alpha,\sigma) = P_x(X_1 > 0, \ldots, X_n > 0),$$0 is the persistence exponent of $$Q_n(x;a,\alpha,\sigma) = P_x(X_1 > 0, \ldots, X_n > 0),$$1 sampled at mesh $$Q_n(x;a,\alpha,\sigma) = P_x(X_1 > 0, \ldots, X_n > 0),$$2. Existence of $$Q_n(x;a,\alpha,\sigma) = P_x(X_1 > 0, \ldots, X_n > 0),$$3 follows from supermultiplicativity via association, and $$Q_n(x;a,\alpha,\sigma) = P_x(X_1 > 0, \ldots, X_n > 0),$$4. Hence

$$Q_n(x;a,\alpha,\sigma) = P_x(X_1 > 0, \ldots, X_n > 0),$$5

so identifying $$Q_n(x;a,\alpha,\sigma) = P_x(X_1 > 0, \ldots, X_n > 0),$$6 is equivalent to a dense-sampling assertion about the jump-driven stable Ornstein–Uhlenbeck process in log-time. In physical time, the sampled rate $$Q_n(x;a,\alpha,\sigma) = P_x(X_1 > 0, \ldots, X_n > 0),$$7 of the $$Q_n(x;a,\alpha,\sigma) = P_x(X_1 > 0, \ldots, X_n > 0),$$8-skeleton converges to $$Q_n(x;a,\alpha,\sigma) = P_x(X_1 > 0, \ldots, X_n > 0),$$9 as $$\Lambda(a,\alpha) = -\lim_{n\to\infty} n^{-1}\log Q_n$$0, and $$\Lambda(a,\alpha) = -\lim_{n\to\infty} n^{-1}\log Q_n$$1; the sharp conjecture is that dense sampling recovers the continuously monitored rate $$\Lambda(a,\alpha) = -\lim_{n\to\infty} n^{-1}\log Q_n$$2.

At $$\Lambda(a,\alpha) = -\lim_{n\to\infty} n^{-1}\log Q_n$$3 the sharp value is settled by existing theory: sign events are scaling-invariant, $$\Lambda(a,\alpha) = -\lim_{n\to\infty} n^{-1}\log Q_n$$4 is a centered stationary Gaussian process with covariance $$\Lambda(a,\alpha) = -\lim_{n\to\infty} n^{-1}\log Q_n$$5 and spectral density proportional to $$\Lambda(a,\alpha) = -\lim_{n\to\infty} n^{-1}\log Q_n$$6, which lies in the class handled by Feldheim–Feldheim–Mukherjee. This gives $$\Lambda(a,\alpha) = -\lim_{n\to\infty} n^{-1}\log Q_n$$7, hence $$\Lambda(a,\alpha) = -\lim_{n\to\infty} n^{-1}\log Q_n$$8 and $$\Lambda(a,\alpha) = -\lim_{n\to\infty} n^{-1}\log Q_n$$9. For $x > 0$0 no analogous theorem applies, since $x > 0$1 is purely jump-driven.

Limitations and open questions

The central open problem is the sharp constant: the paper proves existence of $x > 0$2 for all $x > 0$3 but determines its value only at $x > 0$4. The conjectured value $x > 0$5 would require showing that sub-mesh crossings below zero followed by returns above zero do not alter the first-order rate as $x > 0$6. The authors identify the obstacle concretely: such crossings demand joint control of the mass near zero under the conditioned law, the depth of the first overshoot, and the return probability over the remainder of the mesh interval, uniformly over conditioned states at distance of order $x > 0$7 from zero; an estimate conditioned on a fixed overshoot depth fails because the overshoot can be arbitrarily small. No particular power of $x > 0$8 is asserted. Further questions listed include asymmetric strictly stable innovations, nonzero thresholds (which convert the boundary into a geometrically moving one), subexponential corrections to $x > 0$9, the conditioned law given long survival, and analogues for stochastic PDEs driven by Lévy noise.

Conclusion

The paper establishes that the exponential persistence rate of symmetric stable AR(1) sequences is of exact order $0 < a < 1$00 near the unit root, bounded above by $0 < a < 1$01 via an exact geometric-time embedding into a single stable Lévy path, and bounded below through stable closure under subsampling combined with a monotonicity coupling. Along the way it refutes the regularly-varying-tail specialization of the HKW conjecture for stable laws, and it recasts the identification of the sharp constant as a dense-sampling problem for the stationary stable Ornstein–Uhlenbeck process — solved at $0 < a < 1$02 by Gaussian theory, open for $0 < a < 1$03.