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On the central limit theorem and iterated logarithm law for stationary processes

Published online by Cambridge University Press:  17 April 2009

C.C. Heyde
C.C. Heyde
Affiliation:
Department of Statistics, Faculty of Economics, Australian National University, Canberra, ACT.
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Abstract

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It has recently emerged that a convenient way to establish central limit and iterated logarithm results for processes with stationary increments is to use approximating martingales with stationary increments. Functional forms of the limit results can be obtained via a representation for the increments of the stationary process in terms of stationary martingale differences plus other terms whose sum telescopes and disappears under suitable norming. Results based on the most general form of such a representation are here obtained.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 12 , Issue 1 , February 1975 , pp. 1 - 8
Copyright
Copyright © Australian Mathematical Society 1975

References

[1]Gordin, M.I., “The central limit theorem for stationary processes”, Soviet Math. Dokl. 10 (1969), 1174–1176.Google Scholar
[2]Heyde, C.C., “An iterated logarithm result for autocorrelations of a stationary linear process”, Ann. Probability 2 (1974), 328–332.CrossRefGoogle Scholar
[3]Heyde, C.C. and Scott, D.J., “Invariance principles for the law of the iterated logarithm for martingales and processes with stationary increments”, Ann. Probability 1 (1973), 428–436.Google Scholar
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[5]Scott, D.J., “Central limit theorems for martingales and for processes with stationary increments using a Skorokhod representation approach”, Adv. Appl. Probability 5 (1973), 119–137.Google Scholar