Limiting Behavior of Functionals of the Sample Spectral Distribution

Abstract

The parameters of a stationary process can be viewed as functions of the spectral distribution function. This work concerns (estimators) parameters defined as integrals of $m (\geq 1)$-dimensional kernel functions with respect to the (sample) spectral distribution function. Conditions for asymptotic normality, almost sure convergence, and probability one bounds are derived for such estimators. The approach taken is based upon the reduction of an $m$-dimensional problem to one-dimension via consideration of a Frechet differential and its linearity. The probability one bound for the estimators is obtained by first establishing it $(O((n^{-1}\log n)^{1/2}))$ for the difference of the sample and true spectral distribution functions in the supnorm and then showing that this rate is transferred to the estimators through integration.