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

Abstract

We develop an explicit finite-sample version of the Borel–Cantelli lemma under m-dependence. Given any m-dependent sequence of events (Ak)<inf>1≤k≤N</inf>, we prove P(⋃k=1NA<inf>k</inf>)≥1−exp(−1m+1∑k=1NP(A<inf>k</inf>)). 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 ∑<inf>k=1</inf>^ϕ(n)P(A<inf>k</inf>)≥n for all n≥1, then for every N≥1 and i≥0, P(⋃k=i+1ϕ(i+N)A<inf>k</inf>)≥1−exp(−Nm+1). Finally, we present a complementary second-order refinement involving local pairwise intersection probabilities. These results complement the asymptotic and rate results of Lu et al. (2026) by providing explicit finite-N bounds and a simple comparison framework for the baseline and second-order estimates.