Amarts: A class of asymptotic martingales a. Discrete parameter |
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Authors: | Gerald A. Edgar Louis Sucheston |
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Affiliation: | Department of Mathematics, The Ohio State University, Columbus, Ohio 43210 USA |
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Abstract: | A sequence (Xn) of random variables adapted to an ascending (asc.) sequence n of σ-algebras is an amart iff EXτ converges as τ 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 (EXτ)τ∈T is bounded is uniformly integrable (Theorem 2.9). If Xn is an amart such that supnE(Xn ? Xn?1)2 < ∞, then 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 Zτ is uniformly integrable (Theorem 3.2). If Xn is an asc. amart, τk a sequence of bounded stopping times, k ≤ τk, and E(supk |Xτk ? Xk?1|) < ∞, then there exists a set G such that Xn → a.e. on G and lim inf Xn = ?∞, lim sup Xn = +∞ 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 6Xn6 < ∞}. An asc. or desc. amart converges a.e. weakly if supTE6Xτ6 < ∞ (Theorem 5.2; only the desc. case is new). |
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Keywords: | 60G40 60G45 60G99 60F15 46G10 Amart martingale quasimartingale convergence a.e. Riesz decomposition Doob decomposition law of large numbers weak convergence Radon-Nikodym property |
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