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A computation of the Littlewood exponent of stochastic processes

Published online by Cambridge University Press:  24 October 2008

Ron C. Blei  and
J.-P. Kahane
Ron C. Blei
Affiliation:
Department of Mathematics, University of Connecticut, Storrs, Connecticut 06268, USA.
J.-P. Kahane
Affiliation:
Mathématique, Bát 425, Université de Paris– Sud, 91405 Orsay, France.
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Extract

A stochastic process X = {X(t): t ∈ [0, 1]} on a probability space (Ω, , ℙ) is said to have finite expectation if the function defined on the measureable rectangles in Ω × [0, 1] by

for A ∈ and (s, t) ⊂ [0, 1] gives rise to a complex measure in each of its two coordinates (see [1], definition 1·1). Equivalently, X has finite expectation if

is finite. The function defined by (1), effectively a generalization of the Doléans measure (see e.g. [4] pp. 33–35), is extendible to a bona fide complex measure on Ω × [0, 1] if and only if its ‘total variation’

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 103 , Issue 2 , March 1988 , pp. 367 - 370
Copyright
Copyright © Cambridge Philosophical Society 1988

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References

REFERENCES

[1]Blei, R. C.. Multi-linear measure theory and multiple stochastic integration.Google Scholar
[2]Blei, R. C.. α-Chaos. J. Funct. Anal., to appear.Google Scholar
[3]Blei, R. C. and Körner, T. W.. Combinatorial dimension and random sets. Israel J. of Math. 47 (1984), 65–74.CrossRefGoogle Scholar
[4]Chung, K. L. and Williams, R. J.. Introduction to Stochastic Integration (Birkhauser-Verlag, 1983).CrossRefGoogle Scholar
[5]Littlewood, J. E.. On bounded bilinear forms in an infinite number of variables. Quart. J. Math. Oxford 1 (1930), 164–174.CrossRefGoogle Scholar