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Unformatted text preview: On Strongly Equivalent Nonrandomized Transition Probabilities Eugene A. Feinberg 1 and Alexey B. Piunovskiy 2 Abstract Dvoretzky, Wald and Wolfowitz proved in 1951 the existence of equivalent and strongly equivalent mappings for a given transition probability when the number of nonatomic measures is finite and the decision set is finite. This paper introduces a notion of strongly equivalent transition probabilities with respect to a finite collection of functions. This notion contains the notions of equivalent and strongly equivalent transition probabilities as particular cases. This paper shows that a strongly equivalent map ping with respect to a finite collection of functions exists for a finite number of nonatomic distributions and finite decision set. It also provides a condition when this is true for a countable decision set. Ac cording to a recent example by Loeb and Sun, a strongly equivalent mapping may not exist under these conditions when the decision set is uncountable. This paper also provides two additional counterexam ples and shows that strongly equivalent mappings exist for homogeneous transition probabilities. 1 Introduction Dvoretzky, Wald and Wolfowits [2, 3] studied the following problem. Consider a standard Borel space X , i.e. X is a onetoone measurable image of a Borel subset of a Polish space, and consider another standard Borel space A . Sometimes X is called a state space (or set) and A is called a decision space (or set). Let a finite number of nonatomic probability measures μ 1 ,...,μ K on X be given. We recall that a measure μ on a standard Borel space X is called nonatomic if μ ( { x } ) = 0 for all x ∈ X . We follow the terminology that a finite set is countable. Of course, if A is countable then R A f ( a ) ν ( da ) = ∑ a ∈ A f ( a ) ν ( a ) for any measure ν and for any function f on A . Denote by π ( da  x ) a regular transition probability from X to A . Since we consider only regular transition probabilities and only measurable functions, we sometimes omit the terms “regular” and “measurable.” The following two facts were proved in [3]. Proposition 1. ([3, Theorem 3.1]) If A is a finite set then for any finite number K , for any mea surable functions r k ( x,a ) , k = 1 ,...,K, and for any transition probability π from X to A , there exists a measurable mapping ϕ of X to A such that Z X Z A r k ( x,a ) π ( da  x ) μ k ( dx ) = Z X r k ( x,ϕ ( x )) μ k ( dx ) , k = 1 ,...,K. 1 Department of Applied Mathematics and Statistics, State University of New York at Stony Brook, Stony Brook, NY 11794 3600, USA. 2 Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 7ZL, UK....
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This note was uploaded on 12/06/2011 for the course MATH 101 taught by Professor Eugenea.feinberg during the Fall '11 term at State University of New York.
 Fall '11
 EugeneA.Feinberg

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