Asymptotic Expansions for Sums of Block-Variables under Weak Dependence

Let $\{X_{i}\}_{i=-\infty}^{\infty}$ be a sequence of random vectors and $Y_{in}=f_{in}(\scr{X}_{i},\ell)$ be zero mean block-variables where $\scr{X}_{i,\ell}=(X_{i},\ldots ,X_{i+\ell -1})$, i . 1, are overlapping blocks of length $\ell $ and where $f_{in}$ are Borel measurable functions. This paper establishes valid joint asymptotic expansions of general orders for the joint distribution of the sums $\sum_{i=1}^{n}X_{i}$ and $\sum_{i=1}^{n}Y_{in}$ under weak dependence conditions on the sequence $\{X_{i}\}_{i=-\infty}^{\infty}$ when the block length l grows to infinity. In contrast to the classical Edgeworth expansion results where the terms in the expansions are given by powers of $n^{-1/2}$, the expansions derived here are mixtures of two series, one in powers of $n^{-1/2}$ and the other in powers of $[{\textstyle\frac{n}{\ell}}]^{-1/2}$. Applications of the main results to (i) expansions for Studentized statistics of time series data and (ii) second order correctness of the blocks of blocks bootstrap method are given.