Some Asymptotic Properties of a Two-Dimensional Periodogram.
JOHNS HOPKINS UNIV BALTIMORE MD DEPT OF STATISTICS
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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, the asymptotic finite dimensional distribution functions of the periodogram of an independent, orthonormal field which obeys the Lindeberg-Feller condition are found. This result is extended to cover the periodogram of random fields that may be represented as a moving average of such an orthonormal field. This extension is verified by showing that, save for a scale factor, the asymptotic finite dimensional distribution functions of the two periodograms are equal. The rate of convergence to zero of the mean difference of the two random variables obtained by evaluating the above mentioned periodograms at any fixed point is obtained by paralleling Olshens approach to a similar analysis of the one-dimensional periodogram. The analysis, however, requires defining a certain class of Lipschitz functions in two dimensions, and the derivation of some properties of this class of functions. Author
- Statistics and Probability