## The Annals of Statistics

### Limiting Behavior of Functionals of the Sample Spectral Distribution

Daniel MacRae Keenan

#### 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.

#### Article information

Source
Ann. Statist., Volume 11, Number 4 (1983), 1206-1217.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176346333

Digital Object Identifier
doi:10.1214/aos/1176346333

Mathematical Reviews number (MathSciNet)
MR720265

Zentralblatt MATH identifier
0526.60029

JSTOR