Ziȩba, W. Some special properties of conditional expectation. (English) Zbl 0798.60031 Acta Math. Hung. 62, No. 3-4, 385-393 (1993). 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 W. D. Sudderth [Ann. Math. Stat. 42, 2143-2146 (1971; Zbl 0226.60069)] and A. Engelbert and H. J. Engelbert [Z. Wahrscheinlichkeitstheorie Verw. Geb. 48, 309- 325 (1979; Zbl 0403.60044)]. Some interesting examples involving uniform integrability are also given. Reviewer: R.J.Tomkins (Regina) MSC: 60F15 Strong limit theorems 60G40 Stopping times; optimal stopping problems; gambling theory 60A10 Probabilistic measure theory Keywords:conditional expectation; optimal stopping; finite stopping times; uniform integrability Citations:Zbl 0645.60037; Zbl 0627.60035; Zbl 0226.60069; Zbl 0403.60044 PDFBibTeX XMLCite \textit{W. Ziȩba}, Acta Math. Hung. 62, No. 3--4, 385--393 (1993; Zbl 0798.60031) Full Text: DOI References: [1] D. Blackwell and L. E. Dubins, A converse to the dominated convergence theorem,Illinois Journal of Math.,7 (1963), 508–514. · Zbl 0146.37503 [2] A. Engelbert and H. J. Engelbert, Optimal stopping and almost sure convergence of random sequences.Z. Wahrsch. verw. Gebiete,48 (1979), 309–325. · Zbl 0403.60044 · doi:10.1007/BF00537527 [3] J. Neven.Diskrete-Parameter Martingales, North-Holland Elsevier (1971). [4] W. D. Sudderth, A ”Fatou equation” for randomly stopped variables,The Annals of Math. Stat.,42 (1971), 2143–2146. · Zbl 0226.60069 · doi:10.1214/aoms/1177693082 [5] R. J. Tomkins, On conditional medians,The Annals of Prob.,3 (1975), 375–379. · Zbl 0307.60002 · doi:10.1214/aop/1176996411 [6] W. Zięba, Conditional semiamarts and conditional amarts,Math. Stat. and Prob. Th., vol. A, Proc. of the 6-th Pannonian Symposium on Math. Stat., (1987), pp. 305–315. · Zbl 0632.60041 [7] W. Zięba, A note on conditional Fatou Lemma,Prob. Th. and Related Fields,78 (1987), 73–74. · Zbl 0627.60035 · doi:10.1007/BF00718036 [8] W. Zięba, On theL L f 1 convergence for conditional amarts,J. Multivariate Anal.,26 (1988), 104–110. · Zbl 0649.60040 · doi:10.1016/0047-259X(88)90075-9 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.