Asymptotic expansions for sums of block-variables under weak dependence

Abstract

Let {X_i}_i=−∞^∞ be a sequence of random vectors and $Y_{in}=f_{in}(\mathcal{X}_{i,\ell})$ be zero mean block-variables where $\mathcal{X}_{i,\ell}=(X_{i},\ldots,X_{i+\ell-1})$, i≥1, are overlapping blocks of length ℓ 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 ∑_i=1ⁿX_i and ∑_i=1ⁿY_in under weak dependence conditions on the sequence {X_i}_i=−∞^∞ 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 $[\frac{n}{\ell}]^{-1/2}$. 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.