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.