Finite-sample Borel--Cantelli inequalities under mixing conditions

Abstract

We prove explicit one-lag finite-$N$ lower bounds for $\mathbb P(\bigcup_{k=1}^N A_k)$ using only the marginal probabilities and a selected-lag dependence coefficient of the event-generated $σ$-fields. Each finite statement uses one fixed coefficient convention, either ambient or finite restricted, rather than a limiting mixing assumption. In the $φ$ case, a residue-class blocking argument and a one-sided approximate-independence inequality yield a free spacing parameter $L\ge0$, spacing coefficient $1/(L+1)$, and residual terms governed by $φ(L+1)$. In the $α$ case, we give a simpler one-lag additive-correction bound based on strong-mixing covariance control. We compare these blocking estimates with the Chung--Erdős second-moment bound and with variance criteria for strongly mixing sequences, emphasizing that the $α$-mixing estimate is not intended to replace full-rate variance bounds when the complete decay profile is available. 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 $φ(L+1)=0$ and $α(L+1)=0$ that asymptotically attain the corresponding bound. This sharpness statement does not assert optimality of the residual penalties.