Optional Sampling Sätze

2020 ◽  
pp. 227-237
Author(s):  
Achim Klenke
Keyword(s):  
Optional Sampling

scholarly journals Strong Convergence of Pramarts in Banach Spaces

1981 ◽  
Vol 33 (2) ◽  
pp. 357-361 ◽  
Author(s):  
Leo Egghe
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Probability Space ◽  
Stopping Times ◽  
Good Property ◽  
Convergence Properties ◽  
Good Convergence ◽  
Image Position ◽  
Sampling Property ◽  
Optional Sampling

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 ifconverges 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.,

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