## Department Colloquium

### Friday, October 11th, 2019

**Time:** 2:30 p.m. **Place:** Jeffery Hall 234

**Speaker:** Eugene A. Feinberg (Stony Brook University)

**Title:** Fatou's Lemmas for Varying Probabilities and their Applications to Sequential Decision Making.

**Abstract: ** The classic Fatou lemma states that the lower limit of expectations is greater or equal than the expectation of the lower limit for a sequence of nonnegative random variables. This talk describes several generalizations of this fact including generalizations to converging sequences of probability measures. The three types of convergence of probability measures are considered in this talk: weak convergence, setwise convergence, and convergence in total variation. The talk also describes the Uniform Fatou Lemma (UFL) for sequences of probabilities converging in total variation. The UFL states the necessary and sufficient conditions for the validity of the stronger inequality than the inequality in Fatou's lemma. We shall also discuss applications of these results to sequential optimization problems with completely and partially observable state spaces. In particular, the UFL is useful for proving weak continuity of transition probabilities for posterior state distributions of stochastic sequences with incomplete state observations known under the name of Partially Observable Markov Decision Processes. These transition probabilities are implicitly defined by Bayes' formula, and general method for proving their continuity properties have not been available for long time. This talk is based on joint papers with Pavlo Kasyanov, Yan Liang, Michael Zgurovsky, and Nina Zadoianchuk.

**Prof. Eugene Feinberg** is currently a Distinguished Professor in the Department of Applied Mathematics and Statistics at Stony Brook University. Before coming to Stony Brook, he help positions at Moscow State University of Railway Transportation and Yale. He obtained his Ph.D. from Vilnius University, Lithuania.

Prof. Feinberg is a Fellow of INFORMS and has received several awards including the 2012 IEEE Charles Hirsh Award, the 2012 IBM Faculty Award, and the 2000 Industrial Associates Award from Northrop Grumman.