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

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 (α-)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, err()=-C₁ᵉᶠᶠn-1/2+D()+O(n⁻¹), led by a previously unnoticed variance-inflation floor E_∞()~ c_Φ/ 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 *∝√ n in closed form, with an exponent independent of the dependence; no interior optimum exists when λ≤0 or τ≥1/2. The lattice structure of the pivot also forces any continuous tail approximation to err by at least Θ(m⁻¹-3/2), a bound that is attained. The two main results combine into an interval whose coverage error is O(n⁻¹) near *, turning the block length into a computable quantity; coverage is 95%±0.15% over 16 configurations, against $60.1%$ for the first-order baseline.