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November 2001 Dubins-freedman Processes and RC Filters
Christian Mazza, Didier Piau
Ann. Appl. Probab. 11(4): 1330-1352 (November 2001). DOI: 10.1214/aoap/1015345405

Abstract

We use McFadden’s integral equations for random RC filters to study the average distribution of Dubins–Freedman processes. These distributions are also stationary probability measures of Markov chains on [0,1], defined by the iteration of steps to the left $x \to ux$, and of steps to the right $x \to v + (1 - v)x$, where uand vare random from [0,1]. We establish new algorithms to compute the stationary measure of these chains.

Turning to specific examples, we show that, if the distributions of u and $1-v$ are Beta(a,1), or Beta (a, 2), or if u and $1 - v$ are the exponential of Gamma (a, 2) distributed random variables, then the stationary measure is a combination of various hypergeometric functions, which are often $_3 F_2$ functions. Our methods are based on a link that we establish between these Markov chains and some RC filters. We also determine the stationary distribution of RC filters in specific cases. These results generalize recent examples of Diaconis and Freedman.

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Christian Mazza. Didier Piau. "Dubins-freedman Processes and RC Filters." Ann. Appl. Probab. 11 (4) 1330 - 1352, November 2001. https://doi.org/10.1214/aoap/1015345405

Information

Published: November 2001
First available in Project Euclid: 5 March 2002

zbMATH: 1012.60061
MathSciNet: MR1878300
Digital Object Identifier: 10.1214/aoap/1015345405

Subjects:
Primary: 60F05 , 60J05

Keywords: Dubins–Freedman process , Gauss’s hypergeometric equation , hypergeometric functions , iterated random functions , Random affine system , RC filter

Rights: Copyright © 2001 Institute of Mathematical Statistics

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Vol.11 • No. 4 • November 2001
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