Abstract
This article is concerned with Markov chains on ℝm constructed by randomly choosing an affine map at each stage, and then making the transition from the current point to its image under this map. The distribution of the random affine map can depend on the current point (i.e., state of the chain). Sufficient conditions are given under which this chain is ergodic.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barnsley, M. F., Demko, S. G., Elton, J. H., and Geronimo, J. S. (1988). Invariant measures for Markov process arising from iterated function systems with place-dependent probabilities, Georgia Tech. preprint.
Berger, M. A., and Amit, Y. (1988). Products of random affine maps (to appear).
Brandt, A. (1986). The stochastic equationY n+1=A nYn+B n with stationary coefficients.Adv. Appl. Prob. 18, 211–220.
Furstenberg, H., and Kifer, Y. (1983). Random matrix products and measures on projective space.Israel J. Math. 46, 12–32.
Khas'minskii, R. Z. (1960). Ergodic properties of recurrent diffusion processes and stabilization of the solution to the Cauchy problem for parabolic equations.Theor. Prob. Appl. 5, 179–196.
Khas'minskii, R. Z. (1980).Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Rockville, Maryland (translated from Russian).
Vervaat, W. (1979). On a stochastic difference equation and a representation of non-negative infinitely divisible random variables.Adv. Appl. Prob. 11, 750–783.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Berger, M.A., Soner, H.M. Random walks generated by affine mappings. J Theor Probab 1, 239–254 (1988). https://doi.org/10.1007/BF01246627
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01246627