Let $\{ X_i\}_{ i=-\infty}^{ \infty}$ be a sequence of random vectors and $Y_{ in}=f_{ in}(\mathcal{ X}_{ i,\ell})$ be zero mean block-variables where $\mathcal{ X}_{ i,\ell}=(X_i,...,X_{ i+\ell-1}),i\geq 1$, 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 $\sum_{ i=1}^nX_i$ and $\sum_{ i=1}^nY_{ in}$ under weak dependence conditions on the sequence $\{ X_i\}_{ i=-\infty}^{ \infty}$ when the block length $\ell$ 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.
Categories: math.ST math.PR stat.TH