Bootstrap procedures for AR (∞) — processes

Abstract

In this paper we will deal with an application of Efron’s 1979 bootstrap to stationary stochastic processes in discrete time. In many applications it is assumed that these processes are of autoregressive or more generally of autoregressive moving average type, i.e. the underlying stationary process X = (X _t: t ∈ Z = {0, ±1, ±2,…}) is assumed to satisfy the following stochastic difference equation

$$ {X_t} = \sum\limits_{{v = 1}}^p {{a_v}{X_{{t - v}}} + {\varepsilon_t} + \sum\limits_{{\mu = 1}}^q {{b_{\mu }}{\varepsilon_{{t - \mu }}},\;t \in Z} } $$

Here ε = (ε _t: t ∈ Z) denotes a white noise, that is a sequence of uncorrelated, zero mean random variables with finite variance σ ².