The Annals of Applied Probability

Dubins-freedman Processes and RC Filters

Christian Mazza and Didier Piau

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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.

Article information

Ann. Appl. Probab., Volume 11, Number 4 (2001), 1330-1352.

First available in Project Euclid: 5 March 2002

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Zentralblatt MATH identifier

Primary: 60J05: Discrete-time Markov processes on general state spaces 60F05: Central limit and other weak theorems

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


Mazza, Christian; Piau, Didier. Dubins-freedman Processes and RC Filters. Ann. Appl. Probab. 11 (2001), no. 4, 1330--1352. doi:10.1214/aoap/1015345405.

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