The Annals of Applied Probability

Dubins-freedman Processes and RC Filters

Christian Mazza and Didier Piau

Full-text: Open access

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.

Article information

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

Dates
First available in Project Euclid: 5 March 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1015345405

Digital Object Identifier
doi:10.1214/aoap/1015345405

Mathematical Reviews number (MathSciNet)
MR1878300

Zentralblatt MATH identifier
1012.60061

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

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

Citation

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


Export citation

References

  • [1] Abramowitz, M. and Stegun, I. A. (eds.) (1992). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York. (Reprint of 1972 edition.)
  • [2] Chamayou, J.-F. and Letac, G. (1991). Explicit stationary distributions for compositions of random functions and products of random matrices. J. Theoret. Probab. 4 3-36.
  • [3] Diaconis, P. and Freedman, D. (1999). Iterated random functions. SIAM Rev., 41 45-76.
  • [4] Dubins, L. E. and Freedman, D. A. (1967). Random distribution functions. Proc. Fifth Berkeley Symp. Math. Statist. Probab. 2 183-214. Univ. California Press, Berkeley.
  • [5] McFadden, J. A. (1959). The probability density of the output of an RC filter when the input is a binary random process. IRE Trans. IT-5 174-178.
  • [6] Munford, A. G. (1986). Moments of a filtered binary process. IEEE Trans. Inform. Theory 32 824-826.
  • [7] Pawula, R. F. and Rice, S. O. (1986). On filtered binary processes. IEEE Trans. Inform. Theory 32 63-72.
  • [8] Pawula, R. F. and Rice, S. O. (1987). A differential equation related to a random telegraph wave problem. Computer calculation of series solution. IEEE Trans. Inform. Theory 33 882-888.
  • [9] Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Springer, New York.
  • [10] Whittaker, E. T. and Watson. G. N. (1996). An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions. A Course of Modern Analysis, 4th ed. (Reprint of the 1927 editon.) Cambridge Univ. Press.