### Dubins-freedman Processes and RC Filters

Christian Mazza and Didier Piau
Source: Ann. Appl. Probab. Volume 11, Number 4 (2001), 1330-1352.

#### 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.

First Page:
Primary Subjects: 60J05, 60F05
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.aoap/1015345405
Digital Object Identifier: doi:10.1214/aoap/1015345405
Mathematical Reviews number (MathSciNet): MR1878300
Zentralblatt MATH identifier: 1012.60061

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