Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-24T10:48:21.780Z Has data issue: false hasContentIssue false
  • English
  • Français

Central limit theorems for martingales and for processes with stationary increments using a Skorokhod representation approach

Published online by Cambridge University Press:  01 July 2016

D. J. Scott
D. J. Scott*
Affiliation:
The Australian National University
*
∗Now at La Trobe University, Bundoora, Victoria.
Get access Rights & Permissions [Opens in a new window]

Abstract

The Skorokhod representation for martingales is used to obtain a functional central limit theorem (or invariance principle) for martingales. It is clear from the method of proof that this result may in fact be extended to the case of triangular arrays in which each row is a martingale sequence and the second main result is a functional central limit theorem for such arrays. These results are then used to obtain two functional central limit theorems for processes with stationary ergodic increments following on from the work of Gordin. The first of these theorems extends a result of Billingsley for Φ-mixing sequences.

Keywords

MARTINGALESCENTRAL LIMIT THEOREMINVARIANCE PRINCIPLETRIANGULAR ARRAYSSTATIONARY PROCESSES
Type
Research Article
Information
Advances in Applied Probability , Volume 5 , Issue 1 , April 1973 , pp. 119 - 137
Copyright
Copyright © Applied Probability Trust 1973 

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. (1965) Ergodic Theory and Information. Wiley, New York.Google Scholar
[2] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
[3] Breiman, L. (1968) Probability. Addison-Wesley, Reading, Mass.Google Scholar
[4] Brown, B. M. (1971) Martingale central limit theorems. Ann. Math. Statist. 42, 59–66.CrossRefGoogle Scholar
[5] Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
[6] Dubins, L. E. (1968) On a theorem of Skorokhod. Ann. Math. Statist. 39, 2094–2097.CrossRefGoogle Scholar
[7] Freedman, D. A. (1967) Some invariance principles for functions of a Markov chain. Ann. Math. Statist. 38, 1–7.CrossRefGoogle Scholar
[8] Gordin, M. I. (1969) The central limit theorem for stationary processes. Soviet Math. Dokl. 10, 1174–1176.Google Scholar
[9] Hall, W. J. (1968) On the Skorokhod embedding theorem. Technical Report No. 33, Department of Statistics, Stanford University.Google Scholar
[10] Hannan, E. J. (1970) Multiple Time Series. Wiley, New York.Google Scholar
[11] Heyde, C. C. and Brown, B. M. (1970) On the departure from normality of a certain class of martingales. Ann. Math. Statist. 41, 2161–2165.CrossRefGoogle Scholar
[12] Heyde, C. C. and Seneta, E. (1972) Estimation theory for growth and immigration rates in a multiplicative process. J. Appl. Prob. 9, 235–256.CrossRefGoogle Scholar
[13] Loynes, R. M. (1970) An invariance principle for reversed martingales. Proc. Amer. Math. Soc. 25, 56–64.Google Scholar
[14] Pratt, J. W. (1960) On interchanging limits and integrals. Ann. Math. Statist. 31, 74–77.Google Scholar
[15] Root, D. H. (1969) The existence of certain stopping times on Brownian motion. Ann. Math. Statist. 40, 715–718.CrossRefGoogle Scholar
[16] Simmons, G. F. (1963) Introduction to Topology and Modern Analysis. McGraw-Hill, New York.Google Scholar
[17] Skorokhod, A. V. (1965) Studies in the Theory of Random Processes. Addison-Wesley, Reading, Mass.Google Scholar
[18] Strassen, V. (1964) An invariance principle for the law of the iterated logarithm. Z. Wahrscheinlichkeitstheorie verw. Geb. 3, 211–226.Google Scholar
[19] Strassen, V. (1967) Almost sure behaviour of sums of independent random variables and martingales. Proc. Fifth. Berkeley Symp. Math. Statist. Prob. 2, 315–343, Univ. of California Press.Google Scholar