IJPAM: Volume 25, No. 1 (2005)

APPROXIMATION OF ABSOLUTELY CONTINUOUS
INVARIANT MEASURES FOR MARKOV SWITCHING
POSITION DEPENDENT RANDOM MAPS

Md. Shafiqul Islam$^1$, Pawel Góra$^2$, Abraham Boyarsky$^3$
$^1$Department of Mathematics and Computer Science
Faculty of Arts and Science
University of Lethbridge
4401 University Drive, Lethbridge, Alberta, T1K 3M4, CANADA
e-mail: shafiqul.islam@uleth.ca
$^{2,3}$Department of Mathematics and Statistics
Concordia University
7141 Sherbrooke Street West, Montreal, Quebec, H4B 1R6, CANADA
$^2$e-mail: pgora$@$vax2.concordia.ca
$^3$e-mail: boyar@alcor.concordia.ca


Abstract.A Markov switching position dependent random map is a random map of a finite number of measurable transformations where the probability of switching from one transformation to another is controlled by a position dependent irreducible stochastic matrix $W$. Existence of absolutely continuous invariant measures (acim) for a Markov switching position dependent random map was proved in [#!BGB1!#] using spectral properties of Frobenius-Perron operator and geometric conditions respectively. In this note, we present a bounded variation proof for the existence of absolutely continuous invariant measures and we describe a method of approximating the invariant measures for Markov switching position dependent random maps. The method is known as Ulam's method.

Received: September 7, 2005

AMS Subject Classification: 37M25

Key Words and Phrases: Markov switching position dependent random maps, absolutely continuous invariant measure, Perron-Frobenius operator, Ulam's method of approximation

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 25
Issue: 1