Finite-sample Borel--Cantelli inequalities under mixing conditions

2026/04/30 by Chatchawan Panraksa
Mathematics · #math.PR #msc:60F15 #msc:60G10 #msc:60E15

paper · pdf · doi:10.1016/j.spl.2026.110902

published as Statistics & Probability Letters 239 (2026), 110902 · 9 pages. Revised to match the published version in Statistics & Probability Letters; referee comments addressed

arxiv created 2026/08/04 · arxiv updated 2026/08/05

Abstract

We prove explicit one-lag, finite-N lower bounds for \mathbb P(\bigcup_k=1^NA_k) that use only the marginal probabilities \mathbb P(A_k) and a single selected-lag dependence coefficient of the event-generated σ-fields. A residue-class blocking argument gives, under φ-mixing, a bound with a free spacing parameter L≥ 0, spacing constant 1/(L+1), and residual governed by φ(L+1); a strong-mixing covariance argument gives an α-mixing analogue with an additive residual \lceil N/(L+1)\rceil α(L+1). When the lag-(L+1) coefficient vanishes, both reduce to the finite-sample m-dependent bound of Panraksa (2026), and the spacing constant 1/(L+1) is sharp in this zero-residual sense. A second-order Bonferroni refinement and a worked geometrically φ-mixing example are included. The estimates are non-asymptotic one-lag tools, complementary to second-moment and variance criteria rather than competitors to them.

Citations

• Criteria for Borel-Cantelli lemmas with applications to Markov chains and dynamical systems
• On Bounding the Union Probability Using Partial Weighted Information
• Lower Bounds on the Probability of a Finite Union of Events
• Generalization of the Borel-Cantelli Lemma
• Basic Properties of Strong Mixing Conditions. A Survey and Some Open Questions