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THE DISTRIBUTION OF PERIODOGRAM ORDINATES
Author(s) -
ZhaoGuo Chen,
Hannan E. J.
Publication year - 1980
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/j.1467-9892.1980.tb00301.x
Subject(s) - mathematics , periodogram , empirical distribution function , distribution (mathematics) , series (stratigraphy) , fourier series , convergence (economics) , spectral density , stationary process , characteristic function (probability theory) , function (biology) , statistics , variance (accounting) , f distribution , fourier transform , probability density function , mathematical analysis , probability distribution , paleontology , economics , biology , economic growth , business , accounting , evolutionary biology
. The empirical distribution function of y i = I (ω j )/{2π f (ω j )}, ω j = 2π j / T , where I (ω) is the periodogram for a set of observations from a stationary time series with spectral density f (ω), is shown to converge, almost surely, to the distribution with density exp(‐ x), under appropriate conditions. The same methods are used to prove the convergence, almost surely, of an estimate of the prediction error variance constructed from the I (ω j ) and of the complex empirical distribution function based on the Fourier coefficients.