## The Annals of Statistics

### Asymptotic expansions for sums of block-variables under weak dependence

S. N. Lahiri

#### Abstract

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.

#### Article information

Source
Ann. Statist., Volume 35, Number 3 (2007), 1324-1350.

Dates
First available in Project Euclid: 26 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1185458986

Digital Object Identifier
doi:10.1214/009053607000000190

Mathematical Reviews number (MathSciNet)
MR2341707

Zentralblatt MATH identifier
1132.60025

#### Citation

Lahiri, S. N. Asymptotic expansions for sums of block-variables under weak dependence. Ann. Statist. 35 (2007), no. 3, 1324--1350. doi:10.1214/009053607000000190. https://projecteuclid.org/euclid.aos/1185458986

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