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 $\varphi$-mixing, a bound with a free spacing parameter $L\ge 0$, spacing constant $1/(L+1)$, and residual governed by $\varphi(L+1)$; 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 $\varphi$-mixing example are included. The estimates are non-asymptotic one-lag tools, complementary to second-moment and variance criteria rather than competitors to them.
From: Chatchawan Panraksa [view email]
[v1]
Sun, 26 Apr 2026 16:31:01 UTC (26 KB)
[v2]
Mon, 8 Jun 2026 11:53:13 UTC (35 KB)
[v3]
Tue, 4 Aug 2026 11:37:09 UTC (11 KB)
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