Asymptotic expansions for sums of block-variables under weak dependence
  Let $\{X_i\}_{i=-\infty}^{\infty}$ be a sequence of random vectors and $Y_{in}=f_{in}(\mathcal{X}_{i,\ell})$ 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. 
S. N. Lahiri