Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-05-19T14:26:22.748Z Has data issue: false hasContentIssue false
  • English
  • Français

Lp-convergence of a certain class of product martingales

Published online by Cambridge University Press:  20 January 2009

Stamatis Koumandos
Stamatis Koumandos
Affiliation:
Department of Pure Mathematics, The University of Adelaide, G.P.O. Box 498, Adelaide, 5001, South Australiae-mail address: skoumand@spam.maths.adelaide.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We establish the Kakutani dichotomy property for two generalized Rademacher–Riesz product measures μ, ν that either μ, ν are equivalent measures or they are mutually singular according as a certain series converges or diverges. We further give sufficient conditions so that in the equivalence case the Radon–Nikodym derivative dμ/dν belongs to Lp(v) for all positive real numbers p, by proving that a certain product martingale converges in Lp(v) for p ≧ 1.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 37 , Issue 2 , June 1994 , pp. 243 - 254
Copyright
Copyright © Edinburgh Mathematical Society 1994

References

REFERENCES

1.Brown, G. and Moran, W., On orthogonality of Riesz products. Math. Proc. Cambridge Philos. Soc 76 (1974), 173–181.CrossRefGoogle Scholar
2.Brown, G. and Moran, W., Products of random variables and Kakutani's criterion for orthogonality of product measures, J. London Math. Soc. 10 (1975), 401–405.CrossRefGoogle Scholar
3.Doob, J. L., Stochastic Processes (Wiley, New York, 1953).Google Scholar
4.Kakutani, S., On equivalence of infinite product measures. Ann of Math 49 (1948), 214–224.CrossRefGoogle Scholar
5.Karanikas, C. and Koumandos, S., On a generalized entropy's formula, Results in Math. 18 (1990), 254–263.CrossRefGoogle Scholar
6.Kilmer, S. J., and Saeki, S., On Riesz product measures; mutual absolute continuity and singularity, Ann. Inst. Fourier (Grenoble) 38 (1988), 63–93.CrossRefGoogle Scholar
7.Peyrière, J., Étude de quelques propriétés des produits de Riesz. Ann. Inst. Fourier (Grenoble) 25 (1975), 127–169.CrossRefGoogle Scholar
8.Ritter, G., Unendliche produkte unkorrelierter funktionen auf kompakten abelschen gruppen, Math. Scand. 42 (1978), 251–270.CrossRefGoogle Scholar
9.Ritter, G., On Kakutani's dichotomy theorem for infinite products of not necessarily independent functions, Math. Ann. 239 (1979), 35–53.CrossRefGoogle Scholar
10.Shiryayev, A. N., Probability (Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, Hong Kong, 1984).CrossRefGoogle Scholar
11.Williams, D., Probability with Martingales (Cambridge University Press, 1991).CrossRefGoogle Scholar