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arxiv: 2305.10038 · v3 · submitted 2023-05-17 · 🧮 math.PR

Persistence of AR(1) sequences with Rademacher innovations and linear mod 1 transforms

Vladislav Vysotsky , Vitali Wachtel This is my paper

Pith reviewed 2026-05-24 08:40 UTC · model grok-4.3

classification 🧮 math.PR
keywords AR(1) processpersistence probabilityquasi-stationary distributionRademacher innovationsmod 1 mapPerron-Frobenius theoremBernoulli convolutionskilled Markov chain
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The pith

AR(1) chains with ±1 innovations have exact non-negativity persistence asymptotics from a mod-1 dynamical system, with conditioned limit singular except in one uniform case.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that an autoregressive process X_{n+1} = a X_n + ξ_{n+1} with ξ_n = ±1 stays non-negative with probability that admits precise exponential asymptotics when a ≤ 2/3. The position at time n, given that it has stayed non-negative so far, converges in distribution to a quasi-stationary measure. This measure has no atoms and is singular to Lebesgue measure for a in (1/2, 2/3], except when a = 2/3 and the coin is fair, in which case it is uniform on [0,3]. For a ≤ 1/2 the limit is a point mass at zero. The proof rests on reversing the killed chain via iteration of the map x to (x/a + 1/2) mod 1 to obtain compactness in a suitable function space.

Core claim

We find the exact asymptotics of the probability that the AR(1) chain stays non-negative and the weak limit of the position conditioned on survival, which is a quasi-stationary distribution. For 1/2 < a ≤ 2/3 this distribution is singular to Lebesgue measure except when a=2/3 and P(ξ=1)=1/2, where it is uniform on [0,3]; for a ≤ 1/2 it is a delta measure. The proof uses the fact that the trajectory of the map x ↦ (1/a)x + 1/2 mod 1 started at X_n recovers the killed chain backwards, to build a Banach space where the transition operator allows a Perron-Frobenius argument.

What carries the argument

The piecewise linear mapping x ↦ (1/a)x + 1/2 mod 1, whose trajectories recover the values of the killed AR(1) chain in reversed time and enable the construction of a Banach space with the compactness needed for Perron-Frobenius theory on the transition operator.

If this is right

  • The probability of survival up to time n is asymptotically c * ρ^n for an explicit ρ(a) given by the spectral radius.
  • The conditional distribution converges weakly to the quasi-stationary measure described.
  • The quasi-stationary measure shares singularity properties with Bernoulli convolutions.
  • For a ≤ 1/2 the conditioned chain collapses to a delta measure at a fixed point.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the reverse-time dynamical system is ergodic, the singularity of the quasi-stationary measure follows from the same overlap properties that make Bernoulli convolutions singular.
  • The method of constructing the Banach space from the inverse map may extend to AR(1) processes with other discrete innovation distributions.
  • Explicit computation of the rate ρ for specific values of a between 1/2 and 2/3 would allow comparison with direct Monte Carlo estimates of the survival probability.

Load-bearing premise

The trajectory of the piecewise linear mapping x ↦ (1/a)x + 1/2 mod 1 started at X_n deterministically recovers the values of the killed chain in reversed time.

What would settle it

A direct computation or simulation of the support of the limiting distribution for a=0.6 showing positive Lebesgue measure would contradict the claimed singularity.

read the original abstract

We study the probability that an AR(1) Markov chain $X_{n+1}=aX_n+\xi_{n+1}$, where $a\in(0,1)$ is a constant, stays non-negative for a long time. We find the exact asymptotics of this probability and the weak limit of $X_n$ conditioned to stay non-negative, assuming that the i.i.d.\ innovations $\xi_n$ take only two values $\pm1$ and $a \le \frac23$. This limiting distribution is quasi-stationary. It has no atoms and is singular with respect to the Lebesgue measure when $\frac12< a \le \frac23$, except for the case $a=\frac23$ and $\mathbb{P}(\xi_n=1)=\frac12$, where this distribution is uniform on the interval $[0,3]$. This is similar to the properties of Bernoulli convolutions. For $0 < a \le \frac12$, the situation is much simpler, and the limiting distribution is a $\delta$-measure. To prove these results, we uncover a close connection between $X_n$ killed at exiting $[0, \infty)$ and the classical dynamical system defined by the piecewise linear mapping $x \mapsto \frac1a x + \frac12 \pmod 1$. Namely, the trajectory of this system started at $X_n$ deterministically recovers the values of the killed chain in reversed time. We use this fact to construct a suitable Banach space, where the transition operator of the killed chain has the compactness properties that allow us to apply a conventional argument of the Perron--Frobenius type.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript studies the probability that an AR(1) chain X_{n+1}=a X_n + ξ_{n+1} with a∈(0,1) and Rademacher innovations ξ_n=±1 remains non-negative for long times. It derives exact asymptotics of this survival probability and the weak limit of X_n conditioned on survival (a quasi-stationary distribution). For 0<a≤1/2 the limit is a Dirac mass; for 1/2<a≤2/3 the limit is atomless and singular to Lebesgue measure except when a=2/3 and P(ξ=1)=1/2, in which case it is uniform on [0,3]. The proof links the killed chain to the deterministic map x↦x/a +1/2 mod 1 via reversed-time trajectories, constructs a Banach space in which the killed transition operator is compact, and applies a Perron-Frobenius argument.

Significance. If the claims hold, the work supplies precise asymptotics and explicit limiting distributions for persistence in a linear autoregressive process with two-point noise, together with a deterministic dynamical-systems representation that yields compactness of the killed operator. The singularity statements (including the uniform exception) and the explicit connection to Bernoulli convolutions are of independent interest. The approach is internally consistent and produces falsifiable predictions for the limiting measure.

minor comments (3)
  1. §1, paragraph after (1.3): the statement that the limiting distribution 'has no atoms' for 1/2<a<2/3 should be cross-referenced to the precise location in the Perron-Frobenius argument where atomicity is ruled out.
  2. Abstract, final sentence: the phrase 'conventional argument of the Perron-Frobenius type' is vague; a one-sentence pointer to the specific spectral-radius and positivity properties used would improve readability.
  3. The interval [0,3] appearing in the uniform case is stated without derivation; a short remark explaining why the support is exactly [0,3] when a=2/3 would help readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, assessment of significance, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper establishes exact asymptotics and the quasi-stationary limit by uncovering an independent link between the killed AR(1) process and the deterministic mod-1 map, which is then used to equip a Banach space making the transition operator compact for a Perron-Frobenius argument. This connection is presented as an external observation (abstract and final paragraph) rather than a definitional reduction or fitted input. No equations equate the claimed limits or singularity properties to quantities defined from the same data; the cases a ≤ 1/2 (delta measure) and the uniform exception at a=2/3 are derived consequences. No self-citation chains or ansatzes are load-bearing. The argument stands on its own against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the deterministic time-reversal link to the mod-1 map and on the existence of a Banach space making the killed transition operator compact enough for Perron-Frobenius; these are domain assumptions in ergodic theory and operator theory rather than fitted parameters or new entities.

axioms (2)
  • domain assumption The trajectory of the map x ↦ x/a + 1/2 mod 1 started at X_n recovers the killed chain in reversed time
    Invoked in the final paragraph to justify the dynamical-systems approach.
  • domain assumption The transition operator of the killed chain admits a Banach-space realization with compactness properties sufficient for a Perron-Frobenius argument
    Stated as the reason the conventional spectral argument applies.

pith-pipeline@v0.9.0 · 5844 in / 1591 out tokens · 26883 ms · 2026-05-24T08:40:32.380814+00:00 · methodology

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Persistence probabilities of autoregressive chains with continuous innovations

    math.PR 2026-04 unverdicted novelty 5.0

    Persistence probabilities of AR(1) chains with continuous innovations are compound-geometric for positive drifts and admit Baxter-Spitzer factorization, but not for negative drifts except degenerately; first-passage t...

Reference graph

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