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

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.