Tail-Accuracy Degradation and Optimal Block Length for Block Self-Normalized Quantile Empirical Saddlepoint Inference under Dependence

Abstract

Self-normalization combined with empirical saddlepoint approximation reaches higher-order tail accuracy under independent sampling, but under dependence the degradation is unquantified and the block length lacks a basis. This paper quantifies both for the block self-normalized quantile pivot under strong ($\alpha$-)mixing. Self-normalizing the block sums by their own quadratic form frees the pivot from the marginal distribution, and expanding its cumulants by multilinearity places the tail at the $n^{-1/2}$ rather than the $m^{-1/2}$ scale. The relative tail error obeys an explicit law, $\mathrm{err}(\ell)=-C_1^{\rm eff}n^{-1/2}+D(\ell)+O(n^{-1})$, led by a previously unnoticed variance-inflation floor $E_\infty(\ell)\sim c_\Phi/\ell$ that does not vanish with the number of blocks but converges to a positive limit. Balancing that floor against a skewness transient of the same order gives the optimal block length $\ell^{*}\propto\sqrt n$ in closed form, with an exponent independent of the dependence; no interior optimum exists when $\lambda\le0$ or $\tau\ge1/2$. The lattice structure of the pivot also forces any continuous tail approximation to err by at least $\Theta(m^{-1}\ell^{-3/2})$, a bound that is attained. The two main results combine into an interval whose coverage error is $O(n^{-1})$ near $\ell^{*}$, turning the block length into a computable quantity; coverage is $95\%\pm0.15\%$ over 16 configurations, against $60.1\%$ for the first-order baseline.