Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-30T20:26:08.843Z Has data issue: false hasContentIssue false
  • English
  • Français

Some asymptotic properties of a two-dimensional periodogram

Published online by Cambridge University Press:  14 July 2016

M. Pagano
M. Pagano*
Affiliation:
The Johns Hopkins University
Get access Rights & Permissions [Opens in a new window]

Extract

The two-dimensional periodogram has been proposed as an estimator of the spectral density of a real, homogeneous, random field defined over a regular lattice on the plane. In the present paper, results pertaining to the asymptotic distributional properties of such a periodogram are obtained. These results generalize some of the work of Hannan (1960), Walker (1965) and more directly Olshen (1967a,b) concerning asymptotic theory for the periodogram of a stationary time series. Although extension of asymptotic theory for one-dimensional periodograms to a parallel theory for two-dimensional periodograms is not completely straightforward (one runs into problems akin to the problems encountered in extending the theory of one-dimensional trigonometric series to two dimensions), further extensions to asymptotic theory for p-dimensional periodograms (p > 2) is easily accomplished by an obvious mimicry of the definitions, theorems and proofs for the two-dimensional case. Since the notation required for the p-dimensional case is rather cumbersome, we have chosen to give results only for two-dimensional periodograms.

Type
Short Communications
Information
Journal of Applied Probability , Volume 8 , Issue 4 , December 1971 , pp. 841 - 847
Copyright
Copyright © Applied Probability Trust 1971 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Dieudonne, J. (1960) Foundations of Modern Analysis. Academic Press, New York.Google Scholar
Doob, J. L. (1953) Stochastic Processes. John Wiley and Sons, Inc., New York.Google Scholar
Hannan, E. J. (1960) Time Series Analysis. Methuen, London.Google Scholar
Olshen, R. A. (1967a) Asymptotic Properties of the Periodogram of a Discrete Stationary Process. Ph.D. Thesis. Yale University.CrossRefGoogle Scholar
Olshen, R. A. (1967b) Asymptotic properties of the periodogram of a discrete stationary process. J. Appl. Prob. 4, 508–528.CrossRefGoogle Scholar
Tonelli, L. (1928) Serie Trigonometriche. Nicola Zanichelli, Bologna.Google Scholar
Walker, A. M. (1965) Some asymptotic results for the periodogram of a stationary time series. J. Aust. Math. Soc. V: 1, 107–128 and V: 4, 512.CrossRefGoogle Scholar