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July 2007 Asymptotic expansions for sums of block-variables under weak dependence
S. N. Lahiri
Ann. Statist. 35(3): 1324-1350 (July 2007). DOI: 10.1214/009053607000000190


Let {Xi}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 and where fin are Borel measurable functions. This paper establishes valid joint asymptotic expansions of general orders for the joint distribution of the sums ∑i=1nXi and ∑i=1nYin under weak dependence conditions on the sequence {Xi}i=−∞ when the block length 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.


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S. N. Lahiri. "Asymptotic expansions for sums of block-variables under weak dependence." Ann. Statist. 35 (3) 1324 - 1350, July 2007.


Published: July 2007
First available in Project Euclid: 26 July 2007

zbMATH: 1132.60025
MathSciNet: MR2341707
Digital Object Identifier: 10.1214/009053607000000190

Primary: 60F05
Secondary: 62E20 , 62M10

Keywords: Blocks of blocks bootstrap , Cramér’s condition , Edgeworth expansions , moderate deviation inequality , Moving block bootstrap , second-order correctness , Strong mixing , Studentized statistics

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 3 • July 2007
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