Markov Processes Generated by Random Iterates of Monotone Maps: Theory and Applications

The paper is a review of results on the asymptotic behavior of Markov processes generated by i.i.d. iterates of monotone maps. Of particular importance is the notion of splitting introduced by Dubins and Freedman (1966). Some extensions to more general frameworks are outlined, and, finally, a number of applications are indicated.

Stability in Distribution of Randomly Perturbed Quadratic Maps as Markov Processes

Iteration of randomly chosen quadratic maps defines a Markov process: X_{n+1}=\epsilon_{n+1}X_n(1-X_n), where \epsilon_n are i.i.d. with values in the parameter space [0,4] of quadratic maps F_{\theta}(x)=\theta x(1-x). Its study is of significance as an important Markov model, with applications to problems of optimization under uncertainty arising in economics. In this article a broad criterion is established for positive Harris recurrence of X_n.Comment: Published at http://dx.doi.org/10.1214/105051604000000918 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Random Iterates of Monotone Maps

In this paper we prove the existence, uniqueness and stability of the invariant distribution of a random dynamical system in which the admissible family of laws of motion consists of monotone maps from a closed subset of a finite dimensional Euclidean space into itself.

Stability in Distribution of Randomly Perturbed Quadratic Maps as Markov Processes

Iteration of randomly chosen quadtratic maps defines a Markov process: X[subscript n + 1] = epsilon[subscript n + 1] X[subscript n](1 - X[subscript n]), where epsilon[subscript n] are i.i.d. with values in the parameter space [0, 4] of quadratic maps F[subscript theta](x) = theta*x(1 - x). Its study is of significance not only as an important Markov model, but also for dynamical systems defined by the individual quadratic maps themselves. In this article a broad criterion is established for positive Harris recurrence of X[subscript n], whose invariant probability may be viewed as an approximation to the so-called Kolmogorov measure of a dynamical system.

Markov processes generated by random iterates of monotone maps: Theory and applications

The paper is a review of results on the asymptotic behavior of Markov processes generated by i.i.d. iterates of monotone maps. Of particular importance is the notion of splitting introduced by Dubins and Freedman (1966). Some extensions to more general frameworks are outlined, and, finally, a number of applications are indicated