Finite-sample Borel–Cantelli inequalities under mixing conditions

We prove explicit one-lag, finite-N lower bounds for P(⋃_k=1^NA_k) that use only the marginal probabilities 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 ⌈N/(L+1)⌉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.