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arxiv: 2604.23791 · v1 · submitted 2026-04-26 · 🧮 math.PR

Finite-sample Borel--Cantelli inequalities under mixing conditions

Chatchawan Panraksa This is my paper

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classification 🧮 math.PR
keywords Borel-Cantellimixing conditionsfinite-sample inequalitiesunion probabilityphi-mixingalpha-mixingblocking argumentprobability bounds
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The pith

Explicit finite-N lower bounds for the union probability of events hold under quantitative mixing conditions.

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

The paper proves explicit lower bounds on the probability that at least one event in a finite sequence occurs, when the events satisfy known quantitative ϕ-mixing or α-mixing rates. A residue-class blocking technique spaces the indices to apply one-sided approximate-independence inequalities, producing a main term scaled by 1/(L+1) plus residual corrections governed by the mixing coefficient at distance L+1. This gives finite-sample control on how dependence slows the growth of the union probability compared with the independent case. A reader would care because these bounds supply concrete estimates usable in any fixed-length sample from a dependent process, without requiring N to infinity.

Core claim

Under quantitative ϕ-mixing or α-mixing bounds on the σ-algebras generated by the events, explicit inequalities lower-bound P(∪_{k=1}^N A_k) via a residue-class blocking argument together with a one-sided approximate-independence inequality; the ϕ-mixing version carries a free spacing parameter L ≥ 0, a leading coefficient 1/(L+1), and residual terms controlled by ϕ(L+1). An additive-correction form holds for α-mixing via covariance bounds. The coefficient 1/(L+1) is asymptotically sharp for the zero-residual subclass of L-dependent sequences where ϕ(L+1) = 0 or α(L+1) = 0.

What carries the argument

Residue-class blocking argument combined with a one-sided approximate-independence inequality, controlled by free spacing parameter L ≥ 0 and mixing coefficient ϕ(L+1) or α(L+1).

Load-bearing premise

The event sequence must obey known quantitative decay rates on its ϕ-mixing or α-mixing coefficients so that the residual terms after blocking remain controllable.

What would settle it

Compute the exact union probability for a concrete L-dependent sequence with ϕ(L+1) = 0 and check whether it stays above the derived lower bound for every finite N and every admissible L.

read the original abstract

We prove explicit finite-$N$ lower bounds for $\mathbb P(\bigcup_{k=1}^N A_k)$ when the $\sigma$-algebras generated by an event sequence satisfy quantitative $\varphi$- or $\alpha$-mixing bounds. The main $\varphi$-mixing estimate is obtained by a residue-class blocking argument and a one-sided approximate-independence inequality; it has a free spacing parameter $L\ge0$, spacing coefficient $1/(L+1)$, and residual terms governed by $\varphi(L+1)$. For $\alpha$-mixing families, we derive an additive-correction analogue using strong-mixing covariance control. A windowed rate corollary and a second-order Bonferroni refinement parallel the corresponding $m$-dependent finite-sample results. The coefficient $1/(L+1)$ is sharp as a universal spacing constant only in the zero-residual sense: the full mixing classes contain $L$-dependent block constructions with $\varphi(L+1)=0$ and $\alpha(L+1)=0$ that asymptotically attain the corresponding bound. This sharpness statement does not assert optimality of the residual penalties.

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 / 3 minor

Summary. The paper proves explicit finite-N lower bounds on P(∪_{k=1}^N A_k) for sequences of events whose generated σ-algebras satisfy quantitative φ-mixing or α-mixing bounds. The central φ-mixing result is derived via a residue-class blocking argument together with a one-sided approximate-independence inequality; the bound contains a free spacing parameter L ≥ 0, a leading coefficient 1/(L+1), and residual terms controlled by φ(L+1). Parallel additive-correction bounds are obtained for α-mixing via covariance control. The work also supplies a windowed-rate corollary and a second-order Bonferroni refinement, and states that the coefficient 1/(L+1) is sharp in the zero-residual (L-dependent) case.

Significance. If the derivations are correct, the results supply usable, explicit finite-sample lower bounds for unions under verifiable mixing conditions, extending classical Borel–Cantelli statements to dependent settings with concrete constants and a tunable parameter L. The explicit trade-off between the spacing coefficient and the mixing residuals, together with the sharpness statement restricted to L-dependent families, is a constructive feature that can be directly applied in ergodic theory, time-series analysis, and statistical inference. The parallel treatment of both φ- and α-mixing broadens the range of applicability.

major comments (2)
  1. [§3] §3 (main φ-mixing theorem): the one-sided approximate-independence step after residue-class blocking must be checked to confirm that the lower bound on the union probability is preserved without an unintended sign flip or over-subtraction in the residual terms controlled by φ(L+1).
  2. [§4] §4 (α-mixing analogue): the additive correction derived from strong-mixing covariance bounds should be compared quantitatively with the φ-mixing version to verify that the two families of inequalities are consistent when both mixing coefficients are available.
minor comments (3)
  1. [Abstract] The abstract and introduction should explicitly state the dependence of the bounds on the individual event probabilities p_k = P(A_k), since the classical independent-case lower bound involves these quantities.
  2. [Notation and §5] Notation for the blocking sets and the windowed-rate corollary would benefit from a small diagram or explicit indexing to clarify the residue-class partition.
  3. [§6] A brief numerical example illustrating the improvement over the plain Bonferroni lower bound for a concrete L-dependent sequence would strengthen the sharpness claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive assessment, the recommendation of minor revision, and the careful attention to the proofs of the main theorems. We address each major comment point by point below, confirming the correctness of the derivations while offering clarifications and a quantitative comparison.

read point-by-point responses

  1. Referee: [§3] §3 (main φ-mixing theorem): the one-sided approximate-independence step after residue-class blocking must be checked to confirm that the lower bound on the union probability is preserved without an unintended sign flip or over-subtraction in the residual terms controlled by φ(L+1).

    Authors: We have re-examined the one-sided approximate-independence step following the residue-class blocking in the proof of the main φ-mixing result (Theorem 3.1). The blocking partitions the index set into L+1 residue classes with intra-class separation at least L, so that the mixing coefficient φ(L+1) controls the deviation from independence within each class. The one-sided inequality is applied in the form that yields a lower bound on the union probability: the intersection probabilities are bounded from above by the product of marginals plus a term of size φ(L+1), and this upper bound is subtracted. Consequently the residual appears with a negative sign that preserves the lower estimate; there is no sign flip or over-subtraction. The same directionality is used in the classical Bonferroni lower bound, and the mixing error is controlled conservatively. We are satisfied that the lower bound remains valid and can insert a short clarifying sentence after the statement of the inequality if the referee considers it helpful. revision: partial

  2. Referee: [§4] §4 (α-mixing analogue): the additive correction derived from strong-mixing covariance bounds should be compared quantitatively with the φ-mixing version to verify that the two families of inequalities are consistent when both mixing coefficients are available.

    Authors: We agree that an explicit comparison strengthens the presentation. The φ-mixing bound takes the form (1/(L+1))∑P(A_k) minus a residual of order φ(L+1), while the α-mixing bound replaces the multiplicative spacing factor by an additive correction controlled by the covariance bound |Cov(1_{A_i},1_{A_j})|≤4α(L+1). It is a standard fact that α(n)≤φ(n)/2 for every n. Substituting this relation shows that the α-correction is at most half the size of the corresponding φ-residual (when both coefficients are known). Thus the two families are consistent: the φ-version is sharper whenever φ-mixing holds, yet the α-version remains valid and applies under the weaker α-mixing assumption. We will add a short comparative paragraph at the end of §4 in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation proceeds from external quantitative ϕ- or α-mixing assumptions on the σ-algebras, applies a residue-class blocking construction with free spacing parameter L, and invokes a one-sided approximate-independence inequality to obtain the explicit finite-N lower bound on P(∪ A_k). The coefficient 1/(L+1) and residual terms governed by ϕ(L+1) are direct consequences of the blocking and the supplied mixing rates; they are not fitted to the target probability or renamed as predictions. The sharpness claim is restricted to the zero-residual (L-dependent) subclass of mixing sequences, which is an external property of the mixing classes rather than a self-referential reduction. No self-citation is load-bearing for the central inequality, and the parallel m-dependent corollaries are presented as analogies rather than foundational inputs. The argument is therefore self-contained against the stated mixing hypotheses.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The claim rests on the domain assumption that the sequence obeys quantitative mixing bounds; L is a free user parameter rather than a fitted constant, and no new entities are postulated.

free parameters (1)
  • L
    User-chosen non-negative integer spacing parameter in the residue-class blocking construction.
axioms (1)
  • domain assumption The sigma-algebras generated by the event sequence satisfy quantitative phi-mixing or alpha-mixing bounds.
    Invoked to bound the residual terms after blocking; this is the central modeling assumption.

pith-pipeline@v0.9.0 · 5485 in / 1235 out tokens · 32737 ms · 2026-05-08T05:29:39.331953+00:00 · methodology

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Reference graph

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