On convergence in probability of martingale-like sequences
Consider a Banach space \(E\), a complete probability space on it, an increasing sequence (\({\mathcal A}_{n}\)) (\(n \in {\mathbb N}\)) of subfields and \(X_{n} \in L^{1}(P) {\mathcal A}_{n}\)-measurable. The main theorem proved in the paper is that every Banach space valued \(V\)-game \((X_{n})\) (i.e. \(V\) is a cofinal subset of \({\mathbb N}\) and for every \(\varepsilon > 0\) there exists \(p \in {\mathbb N}\) such that \(P(\|X_{q}(v)-X_{q} \|>\varepsilon)<\varepsilon\) for all \(p \leq q \leq v\), \(q \in {\mathbb N}\), \(v \in V\), where \(X_{q}(v)\) is the conditional expectation of \(X_{v}\) with respect to \({\mathcal A}_{q}\)) with \(\liminf_{v\in V}E(\|X_{v}\|)<\infty\) represents uniquely as \(X_{n}=M_{n}+Y_{n}\), where \(M_{n}\) is a uniformly integrable martingale and \(Y_{n} \rightarrow 0\) in probability. Corollaries are deduced. The proof, which does not rely on the author's paper [Acta Math. Hung. 59, No. 3--4, 273--281 (1992; Zbl 0802.46044)], passes by two lemmas, the first showing that \((X_{n})\) contains a subsequence which is a mil (for every \(\varepsilon > 0\) there exists \(p \in {\mathbb N}\) with \(P(\sup_{p\leq q\leq n}\|X_{q}(n)-X_{q} \|>\varepsilon)<\varepsilon\) for all \(n \geq p\)) and the second showing that if a subsequence of \((X_{n})\) converges to \(0\) in probability, then \((X_{n})\) itself does. The first lemma allows the use of results of \textit{M. Talagrand} [Ann. Probab. 13, 1192--1203 (1985; Zbl 0582.60055)].
scientific article; zbMATH DE number 1604253