FePiPur2009_Mar - On Strongly Equivalent Nonrandomized...

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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 one-to-one 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.

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FePiPur2009_Mar - On Strongly Equivalent Nonrandomized...

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