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WROCŁAW UNIVERSITY
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Contents of PMS, Vol. 20, Fasc. 1,
pages 93 - 120
 

A GENERAL CONTRACTION PRINCIPLE FOR VECTOR-VALUED MARTINGALES

Stefan Geiss

Abstract: We prove a contraction principle for vector-valued martingales of type

|||| sum n |||| |||| |||| |||| sum n |||| || Dixi||LX < cp||1s<uip<nAi(Di)||Lp|| Hixi||LX (1 < p < oo ), i=1 p i=1 1
where X is a Banach space with elements x,...,x ,(D )n < L (Q, P) 1 n ii=1 1 a martingale difference sequence belonging to a certain class, (H )n < L (M, n) ii=1 1 a sequence of independent and symmetric random variables exponential in a certain sense, and A i operators mapping each D i into a non-negative random variable. Moreover, special operators A i are discussed and an application to Banach spaces of Rademacher type a (1 < a < 2) is given.

1991 AMS Mathematics Subject Classification: 46B09, 60G44.

Key words and phrases: Vector-valued martingales, exponential random variables, operators defined on martingales, contraction principle.

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