Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-28T19:45:11.907Z Has data issue: false hasContentIssue false
  • English
  • Français

Gordin's theorem and the periodogram

Published online by Cambridge University Press:  14 July 2016

Holger Rootzén
Holger Rootzén*
Affiliation:
Tekniska Högskolan i Lund, Sweden
Get access Rights & Permissions [Opens in a new window]

Abstract

In 1968 M. I. Gordin proved a very strong central limit theorem for stationary, ergodic sequences by means of approximation with martingales. In the present paper Gordin's theorem is generalized to cover also the periodogram of a stationary sequence, and the restriction of ergodicity is removed. It is noted that known central limit theorems for stationary processes can often be generalized to the periodogram by means of this result.

Keywords

PERIODOGRAMCENTRAL LIMIT THEOREMTIME SERIES ANALYSIS
Type
Short Communications
Information
Journal of Applied Probability , Volume 13 , Issue 2 , June 1976 , pp. 365 - 370
Copyright
Copyright © Applied Probability Trust 1976 

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

[1] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
[2] Eagleson, G. K. (1975a) On Gordin's central limit theorem for stationary processes. J. Appl. Prob. 12, 176–179.Google Scholar
[3] Eagleson, G. K. (1975b) Martingale convergence to mixtures of infinitely divisible laws. Ann. Prob. 3, 557–562.Google Scholar
[4] Gordin, M. I. (1969) The central limit theorem for stationary processes. Dokl. Akad. Nauk SSR 188, 1174–1176.Google Scholar
[5] Hannan, E. J. (1973) Central limit theorems for time series regression. Z. Wahrscheinlichkeitsth. 26, 157–170.Google Scholar
[6] Heyde, C. C. (1974) On the central limit theorem for stationary processes. Z. Wahrscheinlichkeitsth. 30, 315–320.Google Scholar
[7] Heyde, C. C. (1975) On the central limit theorem and iterated logarithm law for stationary processes. Bull. Austral. Math. Soc. 12, 1–8.Google Scholar
[8] Rootzén, H. (1976) Fluctuations of sequences which converge in distribution. Ann. Prob. 4.Google Scholar
[9] Rozanov, Yu. A. (1968) Stationary Random Processes. Holden Day, San Francisco.Google Scholar