  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.
