Some special properties of conditional expectation
Let \(X_ 1, X_ 2, \dots\) be a sequence of random variables defined on a probability space \((\Omega, {\mathcal A}, P)\). The author [Probab. Theory Relat. Fields 78, No. 1, 73-74 (1988; Zbl 0627.60035)] established that, if \(\{X_ n\}\) is uniformly \({\mathcal F}\)-integrable for some sigma-field \({\mathcal F} \subset {\mathcal A}\) and if \(X_ n \to X_ 0\) a.s., then \(E(X_ n \mid {\mathcal F}) \to E(X_ 0 \mid {\mathcal F})\) a.s. and \(E(| X_ 0 | \mid {\mathcal F}) < \infty\) a.s. . It is now shown that \(\{X_ n\}\) is uniformly \({\mathcal F}\)-integrable if \(0 \leq X_ n \to X_ 0\) a.s., \(E(X_ n \mid {\mathcal F}) \to E (X_ 0 \mid {\mathcal F})\) a.s. and \(E(| X_ n | \mid {\mathcal F}) < \infty\) a.s., \(n \geq 1\). It is also shown that, for any stochastic sequence \(\{X_ n, {\mathcal F}_ n,n \geq 1\}\), \(e \lim \sup_{\tau \in T} E(X_ \tau \mid {\mathcal F})\) a.s., where \(T\) is the set of all finite stopping times with respect to \(\{{\mathcal F}_ n,n \geq 1\}\). This generalizes results of \textit{W. D. Sudderth} [Ann. Math. Stat. 42, 2143-2146 (1971; Zbl 0226.60069)] and \textit{A. Engelbert} and \textit{H. J. Engelbert} [Z. Wahrscheinlichkeitstheorie Verw. Geb. 48, 309- 325 (1979; Zbl 0403.60044)]. Some interesting examples involving uniform integrability are also given.
scientific article; zbMATH DE number 575893