## The Annals of Statistics

- Ann. Statist.
- Volume 23, Number 4 (1995), 1331-1349.

### On the Asymptotic Behaviour of the Moving Block Bootstrap for Normalized Sums of Heavy-Tail Random Variables

#### Abstract

This paper studies the performance of the moving block bootstrap procedure for normalized sums of dependent random variables. Suppose that $X_1, X_2,\ldots$ are stationary $\rho$-mixing random variables with $\sum \rho (2^i) < \infty$. Let $T_n = (X_1 + \cdots + X_n - b_n)/a_n$, for some suitable constants $a_n$ and $b_n$, and let $T^\ast_{m,n}$ denote the moving block bootstrap version of $T_n$ based on a bootstrap sample of size $m$. Under certain regularity conditions, it is shown that, for $X_n$'s lying in the domain of partial attraction of certain infinitely divisible distributions, the conditional distribution $\hat{H}_{m,n}$ of $T^\ast_{m,n}$ provides a valid approximation to the distribution of $T_n$ along every weakly convergent subsequence, provided $m = o(n)$ as $n \rightarrow \infty$. On the other hand, for the usual choice of the resample size $m = n, \hat{H}_{n,n}(x)$ is shown to converge to a nondegenerate random limit as given by Athreya (1987) when $T_n$ has a stable limit of order $\alpha, 1 < \alpha < 2$.

#### Article information

**Source**

Ann. Statist., Volume 23, Number 4 (1995), 1331-1349.

**Dates**

First available in Project Euclid: 11 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176324711

**Digital Object Identifier**

doi:10.1214/aos/1176324711

**Mathematical Reviews number (MathSciNet)**

MR1353508

**Zentralblatt MATH identifier**

0841.62037

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62E20: Asymptotic distribution theory

Secondary: 62G05: Estimation 60F05: Central limit and other weak theorems

**Keywords**

Moving block bootstrap stable limit $\rho$-mixing stationary Poisson random measure

#### Citation

Lahiri, S. N. On the Asymptotic Behaviour of the Moving Block Bootstrap for Normalized Sums of Heavy-Tail Random Variables. Ann. Statist. 23 (1995), no. 4, 1331--1349. doi:10.1214/aos/1176324711. https://projecteuclid.org/euclid.aos/1176324711