Geodesic
Strong Convergence of Pramarts in Banach Spaces
Canadian journal of mathematics, Tome 33 (1981) no. 2, pp. 357-361

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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., 
Egghe, Leo. Strong Convergence of Pramarts in Banach Spaces. Canadian journal of mathematics, Tome 33 (1981) no. 2, pp. 357-361. doi: 10.4153/CJM-1981-029-8
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