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Strong Convergence of Pramarts in Banach Spaces

Published online by Cambridge University Press:  20 November 2018

Leo Egghe
Leo Egghe*
Affiliation:
L. U. C-Universitaire Campus, Diepenbeek, Belgium
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Let E be a Banach space and be an adapted sequence on the probability space We denote by T the set of all bounded stopping times with respect to . is called a pramart if

converges to zero in probability, uniformly in τ ≧ σ. The notion of pramart was introduced in [6]. A good property is the optional sampling property (see Theorem 2.4 in [6]). Furthermore the class of pramarts intersects the class of amarts, and every amart is a pramart if and only if dim E < ∞ ([2], see also [4]). Pramarts behave indeed quite differently than amarts. Although the class of pramarts is large, they have good convergence properties as is seen in the next two results of Millet-Sucheston, [6], [7].

THEOREM 1.1. Let be a real-valued pramart of class (d), i.e.,

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 2 , 01 April 1981 , pp. 357 - 361
Copyright
Copyright © Canadian Mathematical Society 1981

References

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