Amarts: A class of asymptotic martingales a. Discrete parameter

Abstract

A sequence ( X n ) of random variables adapted to an ascending (asc.) sequence F n of σ-algebras is an amart iff EX τ converges as τ runs over the set T of bounded stopping times. An analogous definition is given for a descending (desc.) sequence F 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 L 1-boundedness). A desc. amart and an asc. L 1-bounded amart converge a.e. (Theorem 2.3; only the desc. case is new). In the desc. case, an adapted sequence such that ( EX τ ) τ∈ T is bounded is uniformly integrable (Theorem 2.9). If X n is an amart such that sup n E( X n − X n−1 ) 2 < ∞, then X n n converges a.e. (Theorem 3.3). An asc. amart can be written uniquely as Y n + Z n where Y n is a martingale, and Z n → 0 in L 1. Then Z n → 0 a.e. and Z τ is uniformly integrable (Theorem 3.2). If X n is an asc. amart, τ k a sequence of bounded stopping times, k ≤ τ k , and E(sup k | X τ k − X k−1 |) < ∞, then there exists a set G such that X n → a.e. on G and lim inf X n = −∞, lim sup X n = +∞ on G c (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 ‖ X n ‖ < ∞}. An asc. or desc. amart converges a.e. weakly if sup T E‖ X τ ‖ < ∞ (Theorem 5.2; only the desc. case is new).