Hostname: page-component-77c89778f8-5wvtr Total loading time: 0 Render date: 2024-07-20T21:24:10.898Z Has data issue: false hasContentIssue false
  • English
  • Français

Characterizations, stochastic equations, and the Gibbs sampler

Part of: Stochastic analysis Markov processes

Published online by Cambridge University Press:  14 July 2016

Stephen Walker  and
Paul Damien
Stephen Walker*
Affiliation:
Imperial College, London
Paul Damien*
Affiliation:
University of Michigan
*
∗Postal address: Department of Mathematics, Imperial College, 180 Queen's Gate, London SW7 2BZ, UK. Email address: s.walker@ma.ic.ac.uk.
∗∗Postal address: Department of Statistics and Management Science, School of Business, University of Michigan, Ann Arbor, 48109–1234, MI.
Get access Rights & Permissions [Opens in a new window]

Abstract

We obtain characterizations of densities on the real line and provide solutions to stochastic equations using the Gibbs sampler. Particular stochastic equations considered are of the type X =dB(X+C) and X =dBX+C.

Keywords

Irreducible Markov chainsconditional density

MSC classification

Primary: 60H99: None of the above, but in this section
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 3 , September 1999 , pp. 747 - 751
Copyright
Copyright © Applied Probability Trust 1999 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Dufresne, D. (1996). On the stochastic equation ℒ(X)=ℒ[B(X+C)] and a property of gamma distributions. Bernoulli 2, 287–291.Google Scholar
Kotz, S., and Steutel, F. W. (1988). A note on the characterisation of exponential distributions. Statist. Prob. Lett. 6, 201–203.Google Scholar
Lukacs, E. (1955). A characterisation of the gamma distribution. Ann. Math. Statist. 26, 319–324.Google Scholar
Roberts, G. O., and Smith, A. F. M. (1994). Simple conditions for the convergence of the Gibbs sampler and Metropolis–Hastings algorithms. Stoch. Proc. Appl. 49, 207–216.Google Scholar
Smith, A. F. M, and Roberts, G. O. (1993). Bayesian computations via the Gibbs sampler and related Markov chain Monte Carlo methods. J. R. Statist. Soc. B 55, 3–23.Google Scholar
Vervaat, W. (1979). On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Adv. Appl. Prob. 11, 750–783.Google Scholar