Finite-sample Borel--Cantelli inequalities under mixing conditions

Abstract

We prove explicit one-lag, finite-$N$ lower bounds for $\mathbb{P}({\bigcup }_{k=1}^N{A}_k)$ that use only the marginal probabilities $\mathbb{P}(A_k)$ and a single selected-lag dependence coefficient of the event-generated $\sigma$-fields. A residue-class blocking argument gives, under $\phi$-mixing, a bound with a free spacing parameter $L\ge 0$, spacing constant $1/(L+1)$, and residual governed by $\phi (L+1)$; a strong-mixing covariance argument gives an $\alpha$-mixing analogue with an additive residual $\lceil N/(L+1)\rceil \alpha (L+1)$. 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 $\phi$-mixing example are included. The estimates are non-asymptotic one-lag tools, complementary to second-moment and variance criteria rather than competitors to them.

arXiv comment9 pages. Revised to match the published version in Statistics & Probability Letters; referee comments addressed

References in corpus (1)

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