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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
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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 

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References

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