Amarts: A class of asymptotic martingales a. Discrete parameter

A sequence (Xn) of random variables adapted to an ascending (asc.) sequence n of [sigma]-algebras is an amart iff EX[tau] converges as [tau] runs over the set T of bounded stopping times. An analogous definition is given for a descending (desc.) sequence n. A systematic treatment of amarts is given. Some results are: Martingales and quasimartingales are amarts. Supremum and infimum of two amarts are amarts (in the asc. case assuming L1-boundedness). A desc. amart and an asc. L1-bounded amart converge a.e. (Theorem 2.3; only the desc. case is new). In the desc. case, an adapted sequence such that (EX[tau])[tau][set membership, variant]T is bounded is uniformly integrable (Theorem 2.9). If Xn is an amart such that supnE(Xn - Xn-1)2 < [infinity], then Xn/n converges a.e. (Theorem 3.3). An asc. amart can be written uniquely as Yn + Zn where Yn is a martingale, and Zn --> 0 in L1. Then Zn --> 0 a.e. and Z[tau] is uniformly integrable (Theorem 3.2). If Xn is an asc. amart, [tau]k a sequence of bounded stopping times, k <= [tau]k, and E(supk X[tau]k - Xk-1) < [infinity], then there exists a set G such that Xn --> a.e. on G and lim inf Xn = -[infinity], lim sup Xn = +[infinity] on Gc (Theorem 2.7). Let E be a Banach space with the Radon-Nikodym property and separable dual. In the definition of an E-valued amart, Pettis integral is used. A desc. amart converges a.e. on the set {lim sup ||Xn|| < [infinity]}. An asc. or desc. amart converges a.e. weakly if supT E||X[tau]|| < [infinity] (Theorem 5.2; only the desc. case is new).