Amarts: A class of asymptotic martingales a. Discrete parameter

Gerald A. Edgar and Louis Sucheston

Journal of Multivariate Analysis, 1976, vol. 6, issue 2, 193-221

Abstract: 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 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 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||

Keywords: Amart; martingale; quasimartingale; convergence; a.e.; Riesz; decomposition; Doob; decomposition; law; of; large; numbers; weak; convergence; Radon-Nikodym; property (search for similar items in EconPapers)
Date: 1976
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