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Subsequence principles for vector-valued random variables

Published online by Cambridge University Press:  24 October 2008

D. J. H. Garling
D. J. H. Garling
Affiliation:
St John's College, Cambridge
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Extract

1. Introduction. Révész(8) has shown that if (fn) is a sequence of random variables, bounded in L2, there exists a subsequence (fnk) and a random variable f in L2 such that converges almost surely whenever . Komlós(5) has shown that if (fn) is a sequence of random variables, bounded in L1, then there is a subsequence (A*) with the property that the Cesàro averages of any subsequence converge almost surely. Subsequently Chatterji(2) showed that if (fn) is bounded in LP (where 0 < p ≤ 2) then there is a subsequence (gk) = (fnk) and f in Lp such that

almost surely for every sub-subsequence. All of these results are examples of subsequence principles: a sequence of random variables, satisfying an appropriate moment condition, has a subsequence which satisfies some property enjoyed by sequences of independent identically distributed random variables. Recently Aldous(1), using tightness arguments, has shown that for a general class of properties such a subsequence principle holds: in particular, the results listed above are all special cases of Aldous' principal result.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 86 , Issue 2 , September 1979 , pp. 301 - 312
Copyright
Copyright © Cambridge Philosophical Society 1979

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References

(1)Aldous, D. J.Limit theorems for subsequences of arbitrarily-dependent sequences of random variables. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 40 (1977), 59–82.CrossRefGoogle Scholar
(2)Chatterji, S. D.A general strong law. Invent. Math. 9 (1970), 235–245.CrossRefGoogle Scholar
(3)Diestel, J. Geometry of Banach spaces – selected topics. Lecture Notes in Mathematics, no. 485 (Springer-Verlag 1975).Google Scholar
(4)Figiei, T.On the moduli of convexity and smoothness. Studio Math. 56 (1976), 121–155.CrossRefGoogle Scholar
(5)KomlóS, J.A generalization of a problem of Steinhaus. Acta Math. Acad. Sci. Hungar. 18 (1967), 217–229.CrossRefGoogle Scholar
(6)Meyer, P. A. Martingales and stochastic integrals: I. Lecture Notes in Mathematics, no. 284 (Springer-Verlag, 1972).Google Scholar
(7)Pisier, G.Martingales with values in uniformly convex spaces. Israel J. Math. 20 (1975), 326–350.CrossRefGoogle Scholar
(8)Révész, P.On a problem of Steinhaus. Acta Math. Acad. Sci. Hungar. 16 (1965), 310–318.Google Scholar
(9)Suchanek, A. M. On almost sure convergence of averages of subsequences of vector-valued functions. (Preprint.)Google Scholar