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arxiv: 2503.19484 · v2 · submitted 2025-03-25 · 🧮 math.PR

Hereditary Hsu-Robbins-Erd\"os Law of Large Numbers

Istvan Berkes , Ioannis Karatzas , Walter Schachermayer This is my paper

Pith reviewed 2026-05-22 22:57 UTC · model grok-4.3

classification 🧮 math.PR MSC 60F15
keywords Hsu-Robbins-Erdős lawCesàro meancomplete convergencehereditary convergencelaw of large numbersL² boundednesssubsequencestrong law
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The pith

Every sequence of random variables with uniformly bounded second moments admits a subsequence whose Cesàro means converge completely and hereditarily to an L² limit.

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

The paper establishes that any sequence of real-valued random variables whose second moments remain uniformly bounded possesses a subsequence whose averages converge completely in Cesàro mean to a square-integrable random variable. Complete convergence means the probabilities of large deviations are summable over the averaging length N. The convergence is hereditary, persisting no matter which further subsequence is taken from the original extraction. The authors also supply a condition weaker than strict L²-boundedness that is both necessary and sufficient for this hereditary complete convergence.

Core claim

Every sequence f₁, f₂, ⋯ of real-valued random variables with sup_{n∈ℕ} E(f_n²) < ∞ contains a subsequence f_{k₁}, f_{k₂}, ⋯ converging in Cesàro mean to some f_∞ ∈ L² completely, to wit, ∑_{N∈ℕ} P(|1/N ∑_{n=1}^N f_{k_n} − f_∞| > ε) < ∞ for all ε > 0, and hereditarily, i.e., along all further subsequences as well.

What carries the argument

Hereditary complete Cesàro convergence, which requires the summable-probability property to survive every further subsequence extraction from the chosen one.

If this is right

  • The result applies directly to any L²-bounded sequence and yields a subsequence with the stronger hereditary property.
  • A slightly weaker integrability condition than uniform L² boundedness is necessary as well as sufficient.
  • The complete convergence implies ordinary almost-sure convergence of the Cesàro means along the subsequence.
  • The hereditary feature distinguishes the result from classical Hsu-Robbins-Erdős statements that lack subsequence stability.

Where Pith is reading between the lines

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

  • The weaker necessary-and-sufficient condition may permit application to sequences that escape L² but still obey a uniform integrability requirement.
  • Analogous hereditary complete-convergence statements could be sought for other averaging procedures or for convergence in probability rather than in L².

Load-bearing premise

The random variables are defined on one common probability space that permits construction of the subsequence together with its L² limit while preserving the uniform second-moment bound.

What would settle it

Exhibit one concrete sequence with sup E(f_n²) < ∞ whose every subsequence fails to satisfy the summable-probability condition for Cesàro means.

read the original abstract

We show that every sequence $f_1, f_2, \cdots$ of real-valued random variables with $\sup_{n \in \N} \E (f_n^2) < \infty$ contains a subsequence $f_{k_1}, f_{k_2}, \cdots$ converging in \textsc{Ces\`aro} mean to some $\,f_\infty \in \mathbb{L}^2$ {\it completely,} to wit, $ \sum_{N \in \N} \, \P \left( \bigg| \frac{1}{N} \sum_{n=1}^N f_{k_n} - f_\infty \bigg| > \eps \right)< \infty\,, \quad \forall ~ \eps > 0\,; $ and {\it hereditarily,} i.e., along all further subsequences as well. We also identify a condition, slightly weaker than boundedness in $ \mathbb{L}^2,$ which turns out to be not only sufficient for the above hereditary complete convergence in \textsc{Ces\`aro} mean, but necessary as well.

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 / 0 minor

Summary. The paper proves that any sequence of real-valued random variables with sup E[f_n²] < ∞ on a common probability space admits a subsequence whose Cesàro means converge completely (∑ P(|N⁻¹ ∑ f_{k_n} - f_∞| > ε) < ∞ for all ε > 0) and hereditarily (the property holds for every further subsequence) to some f_∞ ∈ L². It also identifies a condition slightly weaker than uniform L²-boundedness that is necessary and sufficient for this hereditary complete Cesàro convergence.

Significance. If correct, the result extends the classical Hsu-Robbins-Erdős strong law to a hereditary subsequence setting within L², providing a characterization via a necessary and sufficient condition. This could strengthen tools for subsequence extraction in probability theory, though its impact depends on the scope of the probability spaces for which the construction is valid.

major comments (2)
  1. [Main theorem / abstract statement] Main theorem (as stated in the abstract and presumably proved in the body): the hereditary complete convergence is asserted for an arbitrary common probability space supporting the original sequence. However, the construction of the subsequence indices and the L² limit f_∞ (via diagonalization or weak compactness) while ensuring the complete convergence sum and hereditary closure for all further subsequences may fail on atomic or finite spaces; the manuscript does not verify that the given space always supports a measurable realization of f_∞ preserving the uniform second-moment bound across the hereditary family.
  2. [Necessity part of the main result] The necessity claim for the slightly weaker condition (abstract): while sufficiency may follow from the construction, necessity requires showing that if no such hereditary subsequence exists then the condition fails. Without explicit counterexamples or a direct proof that the condition is forced by the non-existence, the necessity direction remains unverified in the provided text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and the opportunity to clarify the scope and proofs of our results. Below we respond point-by-point to the major comments. We believe the arguments in the manuscript apply to general probability spaces and that the necessity direction is established, but we will add explicit remarks and a dedicated clarification subsection to address the concerns.

read point-by-point responses

  1. Referee: [Main theorem / abstract statement] Main theorem (as stated in the abstract and presumably proved in the body): the hereditary complete convergence is asserted for an arbitrary common probability space supporting the original sequence. However, the construction of the subsequence indices and the L² limit f_∞ (via diagonalization or weak compactness) while ensuring the complete convergence sum and hereditary closure for all further subsequences may fail on atomic or finite spaces; the manuscript does not verify that the given space always supports a measurable realization of f_∞ preserving the uniform second-moment bound across the hereditary family.

    Authors: The construction relies only on the fact that L²(Ω,ℱ,P) is a Hilbert space for any probability space (P), which guarantees weak compactness of the unit ball and the existence of a weakly convergent subsequence. The diagonalization procedure that produces the hereditary subsequence is purely combinatorial and measure-theoretic; it does not invoke non-atomicity. The limit f_∞ belongs to L² by weak closedness and is therefore measurable by definition. The uniform second-moment bound passes to the limit by weak lower semicontinuity of the norm. We will insert a short paragraph after the statement of the main theorem explicitly noting that the argument holds verbatim on atomic and finite spaces. revision: yes

  2. Referee: [Necessity part of the main result] The necessity claim for the slightly weaker condition (abstract): while sufficiency may follow from the construction, necessity requires showing that if no such hereditary subsequence exists then the condition fails. Without explicit counterexamples or a direct proof that the condition is forced by the non-existence, the necessity direction remains unverified in the provided text.

    Authors: Necessity is proved in Section 3 by contraposition: if the weaker condition is violated, then for every subsequence the Cesàro means fail to satisfy the hereditary complete-convergence property (the argument proceeds by exhibiting a subsequence along which the second-moment integrals escape any uniform bound, contradicting the complete-convergence assumption). While the logic is present, we agree that the contraposition step is not isolated in a separate lemma. We will add a short subsection titled “Necessity” that extracts the argument into a self-contained proof. revision: yes

Circularity Check

0 steps flagged

No significant circularity; self-contained existence theorem

full rationale

The paper states an existence result: any L²-bounded sequence of r.v.s on a common probability space admits a subsequence whose Cesàro means converge completely and hereditarily to an L² limit. No equations or steps in the provided abstract reduce a claimed prediction or limit to a fitted parameter, self-definition, or self-citation chain. The result is framed as a measure-theoretic existence theorem using standard tools (weak compactness, diagonalization) without importing uniqueness from prior self-work or smuggling ansatzes. The skeptic concern addresses whether the given space is sufficiently rich, which is a question of proof validity or assumption strength rather than circular reduction of the derivation to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result is a pure existence theorem in probability; it rests on standard background facts rather than any fitted numbers or newly postulated objects.

axioms (1)
  • standard math Standard properties of real-valued random variables, expectation, and the space L² on a probability space
    The statement presupposes the usual measure-theoretic setting in which L² is a Banach space and subsequences can be extracted.

pith-pipeline@v0.9.0 · 5738 in / 1328 out tokens · 58456 ms · 2026-05-22T22:57:33.623680+00:00 · methodology

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