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WROCŁAW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

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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