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arxiv: 2604.07750 · v1 · submitted 2026-04-09 · 🧮 math.PR

A finite-sample Borel--Cantelli inequality under m-dependence

Chatchawan Panraksa This is my paper

Pith reviewed 2026-05-10 18:12 UTC · model grok-4.3

classification 🧮 math.PR
keywords m-dependenceBorel-Cantelli lemmafinite-sample boundunion probabilitydependent eventsexponential inequality
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The pith

For any finite m-dependent sequence of events, the union probability is at least 1 minus exp of minus the sum of the individual probabilities divided by m plus one.

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

The paper proves an explicit lower bound on the probability that at least one of a finite collection of m-dependent events occurs. It partitions the index set into m+1 residue classes modulo m+1 so that the events inside each class are mutually independent, then applies the elementary inequality 1 minus the product of (1 minus p) is at least 1 minus exp of minus the sum of the p's to each class and combines the results. The resulting bound holds for any finite N and any m and recovers the classical independent-events case when m equals zero. A reader would care because it supplies a concrete, non-asymptotic guarantee on the coverage probability of a dependent collection without needing the sum of probabilities to diverge or the horizon to go to infinity.

Core claim

Given any m-dependent sequence of events (A_k) for 1 less than or equal to k less than or equal to N, the probability of the union from k=1 to N is at least 1 minus exp of minus one over m+1 times the sum of the P(A_k). The proof splits the indices into the m+1 arithmetic progressions with difference m+1; each such subsequence consists of independent events. The product bound for independent events then yields the exponential lower bound on each subsequence, and the overall union probability is at least as large as the bound coming from any one subsequence. The paper also states a windowed corollary that replaces the total sum by a guaranteed growth rate of at least N over any window of phi-

What carries the argument

Partitioning the index set into residue classes modulo m+1, which isolates m+1 mutually independent subsequences to which the independent-events union lower bound can be applied separately.

If this is right

  • When the sum of probabilities over a window of length proportional to N reaches at least N, the union over that window has probability at least 1 minus exp of minus N over m+1, uniformly in the starting position.
  • The bound is strictly weaker than the independent case by the factor 1 over m+1 but remains positive and grows with the total probability mass even when the events are dependent.
  • A second-order refinement that inserts local pairwise intersection probabilities can be used to tighten the estimate when those intersections are known to be small.
  • The finite-N form supplies a direct comparison between the baseline exponential bound and any improved estimate that exploits more dependence structure.

Where Pith is reading between the lines

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

  • The same partitioning technique may extend to obtain non-asymptotic recurrence probabilities for m-dependent Markov chains or moving-average processes.
  • When m is fixed while N grows, the bound implies that the waiting time until the first success is at most order m times the reciprocal of the average success probability, with an explicit tail.
  • The windowed corollary could be paired with ergodic theorems to convert almost-sure growth of partial sums into explicit finite-time coverage guarantees.
  • If m grows slowly with N, the bound quantifies how much extra probability mass is needed to compensate for the stronger dependence.

Load-bearing premise

The sequence must satisfy the definition of m-dependence: events whose indices differ by more than m are independent.

What would settle it

For small fixed m and N, explicitly construct a joint distribution on the events that satisfies m-dependence and compute the exact union probability; if that number lies strictly below 1 minus exp of minus the sum of the marginals divided by m+1, the claimed inequality fails.

read the original abstract

We prove an explicit finite-sample version of the Borel--Cantelli lemma under $m$-dependence. Given any $m$-dependent sequence of events $(A_k)_{1\leq k\leq N}$, we show that \[ \mathbb{P}\Bigl(\bigcup_{k=1}^N A_k\Bigr) \ge 1 - \exp\Bigl(-\frac{1}{m+1} \sum_{k=1}^{N} \mathbb{P}(A_k)\Bigr). \] The proof splits the index set into residue classes modulo $m+1$, so that each class consists of mutually independent events, and then applies an elementary product--to--exponential bound. We further derive a quantitative windowed corollary: if the partial sums satisfy \(\sum_{k=1}^{\phi(n)}\mathbb{P}(A_k)\ge n\) for all \(n\ge1\), then for every \(N\ge1\) and \(i\ge0\), \[ \mathbb{P}\Bigl(\bigcup_{k=i+1}^{\phi(i+N)} A_k\Bigr) \ge 1-\exp\Bigl(-\frac{N}{m+1}\Bigr). \] Finally, we present a complementary second-order refinement involving local pairwise intersection probabilities. These results complement the asymptotic and rate results of Lu, Shi and Zhao (2026) by providing explicit finite-$N$ bounds and a simple comparison framework for the baseline and second-order estimates.

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 establishes an explicit finite-sample lower bound for the union probability of an m-dependent sequence of events: for any m-dependent (A_k)_{1≤k≤N}, P(∪_{k=1}^N A_k) ≥ 1 − exp(−1/(m+1) ∑ P(A_k)). The proof partitions the indices into m+1 residue classes modulo m+1 (each class mutually independent by the m-dependence definition), applies the elementary inequality 1−∏(1−p_i)≥1−exp(−∑p_i) within the heaviest class, and obtains the bound by averaging. A windowed corollary and a second-order refinement using local pairwise intersections are also derived, complementing the asymptotic results of Lu, Shi and Zhao (2026).

Significance. The result supplies a simple, parameter-free, finite-N lower bound that is directly usable in applications involving m-dependent processes (e.g., time-series extremes or spatial statistics). The proof relies only on the definition of m-dependence and two universal inequalities, with no fitted constants or hidden restrictions; this explicitness and the accompanying windowed corollary are genuine strengths for quantitative work where asymptotic statements are insufficient.

minor comments (2)
  1. §2, after the statement of the main theorem: the sentence “the heaviest class has sum p_i ≥ (total sum)/(m+1)” would benefit from an explicit sentence noting that the partition is equitable (each class has size ⌈N/(m+1)⌉ or ⌊N/(m+1)⌋), even though the inequality direction is unaffected.
  2. The second-order refinement (final section) is stated without a numbered theorem or displayed inequality; adding a displayed statement parallel to the main theorem would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, accurate summary of the main result, and recommendation to accept the manuscript. We are pleased that the referee highlights the explicitness of the finite-sample bound, its parameter-free nature, and the utility of the windowed corollary for applications.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation partitions the indices into m+1 residue classes modulo m+1 so that events within each class are mutually independent by the given definition of m-dependence. It then invokes the elementary inequality 1 - ∏(1-p_i) ≥ 1 - exp(-∑p_i) on the class whose probability sum is at least the total sum divided by (m+1), and notes that the full union contains any sub-union. Both steps rest only on the m-dependence definition and two universal inequalities with no fitted constants, no self-referential predictions, and no load-bearing self-citations. The windowed corollary and second-order refinement follow by the same direct argument. The proof is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The result rests on the standard definition of m-dependence together with two elementary inequalities that hold for all probabilities; no free parameters are introduced and no new entities are postulated.

axioms (3)
  • domain assumption m-dependence: sigma-algebras separated by more than m indices are independent
    Invoked to guarantee that each residue class modulo m+1 consists of mutually independent events.
  • standard math For any collection of independent events, P(none occur) = product (1 − P(A_i))
    Used after the residue-class partition.
  • standard math 1 − x ≤ exp(−x) for x ∈ [0,1]
    Converts the product bound into the stated exponential form.

pith-pipeline@v0.9.0 · 5566 in / 1362 out tokens · 52660 ms · 2026-05-10T18:12:03.601679+00:00 · methodology

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Works this paper leans on

7 extracted references · 7 canonical work pages

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