# Smoothed periodogram asymptotics and estimation for processes and fields with possible long-range dependence

*C. C. Heyde* and
*R. Gay*

*Stochastic Processes and their Applications*, 1993, vol. 45, issue 1, 169-182

**Abstract:**
In this paper we establish central limit theorems for the smoothed unbiased periodogram [integral operator][pi]-[pi]...[integral operator][pi]-[pi]g([omega],[theta]){I*T,X([omega])-EI*T,X([omega])}d[omega]1...d[omega]r, where {Xt} is a stationary r-dimensional random process or random field, possibly with long-range dependence, which is not necessarily Gaussian. Here I*T,X([omega]) is the unbiased periodogram and g([omega],[theta]) is a smoothing function satisfying modest regularity conditions. This result implies asymptotic normality of the asymptotic quasi-likelihood estimator of a distributional characteristic [theta] of the process {Xt} under very general conditions. In particular, these results show the asymptotic optimality of the Whittle estimation procedure for both short and long-range dependence in the absence of the Gaussian assumption, and extend those of Giraitis and Surgailis (1990) for the case r = 1.

**Date:** 1993

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