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 $\alpha, 1 < \alpha < 2$.