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arxiv: 2604.05670 · v1 · submitted 2026-04-07 · 🧮 math.PR

Persistence probabilities of autoregressive chains with continuous innovations

Titouan Donnart , Thomas Simon This is my paper

Pith reviewed 2026-05-10 19:01 UTC · model grok-4.3

classification 🧮 math.PR
keywords persistence probabilitiesautoregressive chainsBaxter-Spitzer factorizationVan Dantzig problemfirst passage timeslog-concave distributionscompound-geometric distributions
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The pith

For autoregressive chains with positive drifts and continuous innovations, persistence probabilities are compound-geometric and satisfy a Baxter-Spitzer factorization that generalizes the random-walk case.

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

The paper studies persistence probabilities for an AR(1) process driven by continuous innovations. When the drift is positive these probabilities are shown to be compound-geometric and to obey a Baxter-Spitzer factorization. When the drift is negative the same factorization is obstructed by a discrete Van Dantzig problem except in a degenerate case. Under log-concave innovations and positive drift the first-passage time below zero is log-convex; the opposite drift and log-convex innovations yield a log-concave passage-time distribution. The bi-exponential innovation case is worked out explicitly and produces an additive factorization of the exponential law.

Core claim

In the positive-drift regime the persistence probabilities of the AR(1) chain are compound-geometric and obey a Baxter-Spitzer factorization; in the negative-drift regime a discrete analogue of Van Dantzig’s problem appears and prevents the factorization except in degenerate cases. When innovations are log-concave (positive drift) or log-convex (negative drift) the first-passage time below zero inherits the opposite log-concavity property. The bi-exponential case further decomposes into an additive factorization of the exponential distribution.

What carries the argument

The Baxter-Spitzer factorization applied to the generating function of the persistence probabilities, together with the discrete Van Dantzig problem that obstructs it for negative drifts.

If this is right

  • Persistence probabilities admit an explicit compound-geometric representation when the drift is positive.
  • The first-passage time into the negative half-line has a log-convex distribution under log-concave innovations and positive drift.
  • For negative drift and log-convex innovations the first-passage distribution is log-concave.
  • Bi-exponential innovations permit an additive factorization of the exponential law for positive drifts.

Where Pith is reading between the lines

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

  • The sharp contrast between positive and negative drifts may reflect a deeper asymmetry in the renewal structure of autoregressive processes.
  • The discrete Van Dantzig problem identified here could appear in other Markovian persistence settings with continuous state space.
  • Numerical checks of the first-passage distributions for concrete log-concave or log-convex innovations could test the predicted convexity properties directly.

Load-bearing premise

The innovations are continuous random variables whose distribution satisfies either log-concavity or log-convexity on the positive half-line, and the drift is strictly positive or strictly negative.

What would settle it

A counter-example consisting of a continuous innovation distribution with positive drift whose persistence probabilities are not compound-geometric, or a non-degenerate negative-drift example where the Baxter-Spitzer factorization still holds, would falsify the main claims.

read the original abstract

We consider the persistence probabilities of an autoregressive chain of order one with continuous innovations. In the case of positive drifts, we show that these persistence probabilities are compound-geometric and satisfy a Baxter-Spitzer factorization generalizing that of the random walk. In the case of negative drifts, we exhibit a discrete Van Dantzig problem, which implies that the Baxter-Spitzer factorization never happens, except in a degenerate case. For positive drifts and log-concave innovations, we show that the first passage time in $(-\infty,0)$ has a log-convex distribution, whereas in the case of negative drifts and log-convex innovations on ${\mathbb R}^+$, it has a log-concave distribution. The case of the bi-exponential innovations is studied in detail, which leads for positive drifts to an additive factorization of the exponential law.

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 / 2 minor

Summary. The paper examines persistence probabilities for autoregressive chains of order one with continuous innovations. For positive drifts, it proves that these probabilities are compound-geometric and obey a Baxter-Spitzer factorization that generalizes the random walk case. For negative drifts, it constructs a discrete Van Dantzig problem showing that the factorization fails except in degenerate cases. Additional results include log-convexity of the first passage time distribution for positive drifts with log-concave innovations and log-concavity for negative drifts with log-convex innovations on the positive reals. The bi-exponential innovation distribution is treated in detail, resulting in an additive factorization of the exponential law for positive drifts.

Significance. If the derivations hold, the results provide a significant extension of fluctuation theory to dependent processes, specifically AR(1) chains. The compound-geometric characterization and factorization for positive drifts, contrasted with the non-factorization for negative drifts via the Van Dantzig analogy, clarify the role of drift sign in solvability. The log-concavity/convexity results on passage times and the explicit bi-exponential example add concrete value and verifiability to the theoretical claims.

minor comments (2)
  1. [Abstract] The abstract introduces the 'discrete Van Dantzig problem' without a brief definition or reference; a short explanatory clause or citation would improve readability for readers unfamiliar with the classical Van Dantzig problem.
  2. [Introduction] The precise support assumptions on the innovation distribution (e.g., whether the density is defined on all of R or restricted to R+) are stated only implicitly in the results; an explicit summary of these conditions in the introduction would strengthen the setup.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment and accurate summary of our results on persistence probabilities for AR(1) chains. The report correctly identifies the compound-geometric structure and Baxter-Spitzer factorization for positive drifts, the counterexample via discrete Van Dantzig for negative drifts, the log-convexity/concavity properties of first-passage times, and the detailed bi-exponential case. No specific major comments or requests for changes were raised.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives persistence probabilities for AR(1) chains with continuous innovations, establishing compound-geometric forms and Baxter-Spitzer factorizations under positive drift (via probabilistic identities) and non-factorization under negative drift (via a discrete Van Dantzig problem). These rest on explicit assumptions of continuity, strict drift sign, and log-concavity/convexity of the innovation density, with the bi-exponential case recovered as a special instance. No derivation step reduces by construction to a fitted input, self-definition, or load-bearing self-citation; the claims are self-contained distributional results independent of the target quantities.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on the standard probabilistic setup of an AR(1) recursion driven by i.i.d. continuous innovations whose distribution is either log-concave or log-convex; no free parameters or new invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Innovations are i.i.d. continuous random variables.
    Explicitly stated in the title and abstract as the driving noise of the autoregressive chain.
  • domain assumption The drift parameter is strictly positive or strictly negative.
    The two main cases are separated by the sign of the drift; zero-drift behavior is excluded.

pith-pipeline@v0.9.0 · 5433 in / 1529 out tokens · 65150 ms · 2026-05-10T19:01:06.855438+00:00 · methodology

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Reference graph

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