Some remarks on pramarts and mils

Zhen-Peng Wang, Xing-Hong Xue
1993 Glasgow Mathematical Journal  
Notations and summary. Let F be a Banach space, (Q, &, P) a fixed probability space, D a directed set filtering to the right with the order ^ , and (&*" D) a stochastic basis of %F, i.e. ($;, D) is an increasing family of sub-cr-algebras of ^:^< =^ for any s,t e D and s s i . Throughout this paper, (A*,) is an F-valued, ($;)-adapted sequence, i.e. X, is ^-measurable, teD. We also assume that X, e L\ i.e. J \\X,\\ < °°. We use I{H) to denote the indicator function of an event H. Let °o be a such
more » ... H. Let °o be a such element: t<°°, teD, D = D U°°, and $L, = a( U ^) -A stopping time is a map r:Q-»D such that (r<38 and %^y. If (^,) satisfies the Vitali condition V, particularly, if D = N = {1,2,. . .}, then (X,) e % if and only if (X,) e y (cf. [18], [23], and [20]). Hence, in this case, if=%. Mucci ([21], [22]) and Millet and Sucheston ([19], [20]) introduced the notations of martingales in the limit, pramarts, and subpramarts, generalizing those of martingales, amarts (Edgar and Sucheston [7]), uniform amarts (Bellow [2]), and submartingales, and provided some sufficient conditions to ensure that (X,) e SP (cf. monographs [10] and [15]). DEFINITION 1 ([21], [20]). A stochastic process (X" 9>" D) is called a martingale in the limit if ess lim ess sup H^, -E(X S \ $j)|| = 0 a.s.
doi:10.1017/s0017089500009800 fatcat:j77fm7v6wvhjlezshr4jqik74e