Convergence rates in the central limit theorem for means of autoregressive and moving average sequences
type: Item
instance of: scholarly article
title: Convergence rates in the central limit theorem for means of autoregressive and moving average sequences
zbMATH Open document ID: 0762.60018
DOI: 10.1016/0304-4149(92)90079-6
published in: Stochastic Processes and their Applications
publication date: 1993-01-17T00:00:00Z
review text: Let an autoregression be given by \(\sum^ \infty_{k=0} b_ k X_{j- k}=\xi_ j\), \(-\infty<j<\infty\), where the innovations \(\xi_ j\) are i.i.d. random variables with zero mean and unit variance (the \(b_ k\)'s being constants). Assume that the function \(B(\theta)=\sum^ \infty_{k=0} b_ k\cos(k\theta)\), \(0<\theta<2\pi\), is well defined and satisfies \(\int^{2\pi}_ 0 B(\theta)^{-2} d\theta<\infty\). Then \(X_ j=\sum^ \infty_{k=0} a_ k\xi_{j-k}\), \(-\infty<j<\infty\). Suppose that the covariance function is regularly varying with exponent \(-\alpha\) \((\alpha\geq 0)\). It is shown that the rate of convergence in a CLT for \(\overline X={1\over n}(X_ 1+\cdots+X_ n)\) is identical to that in the CLT for \(\overline\xi={1\over n}(\xi_ 1+\cdots+\xi_ n)\) iff \(\alpha>0\). Strikingly, when \(\alpha=0\), the corresponding rate for \(\overline X\) can be faster than that for \(\overline\xi\); it can, however, never be slower.
reviewed by: Klaus Schürger
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G10
Mathematics Subject Classification ID: 62M10
zbMATH DE Number: 98447
zbMATH Keywords: central limit theorem for means of autoregressive and moving average sequences
zbMATH Keywords: covariance function
zbMATH Keywords: rate of convergence
MaRDI profile type: Publication
full work available at URL: 0304-4149(92)90079-6
OpenAlex ID: W1973762403
cites work: Q3738326
cites work: Q4726487
cites work: Limit theory for moving averages of random variables with regularly varying tail probabilities
cites work: Q3727187
cites work: The Invariance Principle for Stationary Processes
cites work: Q3962219
cites work: Approximations to Some Finite Sample Distributions Associated with a First-Order Stochastic Difference Equation
cites work: Edgeworth and saddlepoint approximations in the first-order noncircular autoregression
cites work: On the second order asymptotic efficiency of estimators of Gaussian ARMA processes
cites work: On the asymptotic behavior of trigonometric series. I
cites work: On the asymptotic behavior of trigonometric series. II
author: Peter Hall
description: scientific article; zbMATH DE number 98447