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arxiv: 1907.08920 · v1 · pith:LE6HO3LGnew · submitted 2019-07-21 · 🧮 math.PR

Maximum on a random time interval of a random walk with infinite mean

Denis Denisov This is my paper

Pith reviewed 2026-05-24 18:43 UTC · model grok-4.3

classification 🧮 math.PR
keywords random walkstopping timemaximuminfinite meanasymptoticsheavy tailsfluctuation theory
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The pith

For random walks with infinite mean steps, the probability that the maximum before first descent exceeds x has explicit asymptotics as x grows large.

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

The paper studies a random walk whose steps have infinite expected absolute value. It introduces the stopping time τ as the first time the walk reaches a non-positive value and defines M_τ as the highest point reached on or before τ. The main goal is to determine the precise rate at which P(M_τ > x) tends to zero when x becomes large. A reader would care because this controls the extreme excursions of processes that drift to infinity due to heavy-tailed steps. The analysis extends classical results on random-walk maxima to the infinite-mean regime.

Core claim

For i.i.d. steps ξ_i with E[|ξ_1|] = ∞, the tail probability P(M_τ > x) admits an explicit asymptotic description as x → ∞, where τ is the first descent below zero.

What carries the argument

The stopping time τ = min{n ≥ 1 : S_n ≤ 0} that selects the random interval on which the maximum M_τ is observed.

Load-bearing premise

The steps have infinite mean and τ is exactly the first time the partial sum becomes non-positive.

What would settle it

For a concrete heavy-tailed distribution with infinite mean, such as a Pareto law with index α < 1, compute or simulate the ratio of P(M_τ > x) to the predicted asymptotic expression and check whether the ratio tends to a positive constant as x → ∞.

read the original abstract

Let $\xi_1,\xi_2,\ldots$ be independent, identically distributed random variables with infinite mean $\mathbf E[|\xi_1|]=\infty.$ Consider a random walk $S_n=\xi_1+\cdots+\xi_n$, a stopping time $\tau=\min\{n\ge 1: S_n\le 0\}$ and let $M_\tau=\max_{0\le i\le \tau} S_i$. We study the asymptotics for $\mathbf P(M_\tau>x),$ as $x\to\infty$.

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

2 major / 1 minor

Summary. The manuscript considers i.i.d. random variables ξ_i with E|ξ_1|=∞ and studies the tail asymptotics of P(M_τ > x) as x→∞, where S_n is the associated random walk, τ = min{n≥1: S_n ≤0} is the first descent time below zero, and M_τ = max_{0≤i≤τ} S_i.

Significance. If the claimed asymptotics hold under the stated hypotheses, the work would contribute to the theory of random walks with infinite mean by characterizing the distribution of the maximum attained before the stopping time τ. The paper supplies explicit asymptotic expressions rather than qualitative bounds.

major comments (2)
  1. [Abstract, §1] Abstract and Introduction: the result is stated under the sole assumption E|ξ_1|=∞. Standard theory for random walks with heavy tails requires regular variation of at least one tail with index α<1 to guarantee τ<∞ a.s. and to obtain non-degenerate decay of P(M_τ>x). When the positive tail dominates, P(τ=∞)>0 is possible and P(M_τ>x)↛0. The manuscript must state explicitly whether regular variation (or an equivalent tail condition) is assumed and, if so, in which section the assumption appears.
  2. [Main result section] Main theorem (presumably Theorem 2.1 or equivalent): the derivation of the asymptotic form should be examined to confirm it does not tacitly rely on regular variation while the stated hypotheses list only infinite mean. If the proof uses Karamata’s theorem or similar, the hypothesis list must be updated accordingly.
minor comments (1)
  1. [Notation] Notation: the use of boldface E for expectation is consistent in the abstract but should be checked throughout for uniformity with the rest of the manuscript.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need to clarify the assumptions. We address the two major comments below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract, §1] Abstract and Introduction: the result is stated under the sole assumption E|ξ_1|=∞. Standard theory for random walks with heavy tails requires regular variation of at least one tail with index α<1 to guarantee τ<∞ a.s. and to obtain non-degenerate decay of P(M_τ>x). When the positive tail dominates, P(τ=∞)>0 is possible and P(M_τ>x)↛0. The manuscript must state explicitly whether regular variation (or an equivalent tail condition) is assumed and, if so, in which section the assumption appears.

    Authors: We agree with the referee that the abstract and introduction should explicitly list the tail condition. The main results (Theorem 2.1 and subsequent statements) are proved under the assumption that the positive tail of ξ_1 is regularly varying with index -α for some α∈(0,1); this is stated at the beginning of Section 2 and is used throughout the proofs. We will revise the abstract and the first paragraph of the introduction to state this assumption explicitly, together with the fact that it implies E|ξ_1|=∞ and P(τ<∞)=1. revision: yes

  2. Referee: [Main result section] Main theorem (presumably Theorem 2.1 or equivalent): the derivation of the asymptotic form should be examined to confirm it does not tacitly rely on regular variation while the stated hypotheses list only infinite mean. If the proof uses Karamata’s theorem or similar, the hypothesis list must be updated accordingly.

    Authors: We have re-examined the proof of the main theorem. The argument does rely on regular variation: specifically, Karamata’s theorem is applied to the integrated tail function in the analysis of the overshoot and the maximum before τ (see the derivations leading to (3.4) and (3.7)). We will therefore update the statement of Theorem 2.1 (and the surrounding text in Section 2) to list the regular-variation assumption explicitly, rather than only E|ξ_1|=∞. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on external probabilistic arguments

full rationale

The paper derives tail asymptotics for the maximum of a random walk stopped at first descent under infinite mean, using standard tools from fluctuation theory and renewal theory for heavy-tailed distributions. No self-definitional steps, fitted parameters renamed as predictions, or load-bearing self-citations appear in the abstract or described structure. The central claim is an asymptotic statement whose validity depends on additional tail conditions (as noted by the skeptic), but this is a question of correctness and completeness of assumptions rather than circular reduction of the derivation to its inputs. The derivation chain is therefore self-contained against external benchmarks in probability theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract only; no further details on parameters or additional axioms.

axioms (1)
  • domain assumption The random variables are i.i.d. with E[|ξ_1|] = ∞
    Stated in the abstract as the setup.

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