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MATHEMATICAL MODELING AND NUMERICAL SIMULATION
Representation of an invariant measure of irreducible discrete-timemarkov chain with a finite state space by a set of opposite directed trees
A. L. Krugly Scientific Research Institute for System Analysis of the Russian Academy of Science, 36, k. 1, Nahimovskiy pr., Moscow, 117218, Russia
Abstract:
A problem of finding of an invariant measure of irreducible discrete-time Markov chain with a finite state space is considered. There is a unique invariant measure for such Markov chain that can be multiplied by an arbitrary constant. A representation of a Markov chain by a directed graph is considered. Each state is represented by a vertex, and each conditional transition probability is represented by a directed edge. It is proved that an invariant measure for each state is a sum of $n^{n-2}$ non-negative summands, where $n$ is a cardinality of state space. Each summand is a product of $n-1$ conditional transition probabilities and is represented by an opposite directed tree that includes all vertices. The root represents the considered state. The edges are directed to the root. This result leads to methods of analyses and calculation of an invariant measure that is based on a graph theory.
Keywords:
Markov chain, invariant measure, directed tree.
Received: 06.07.2014 Revised: 20.02.2015
Citation:
A. L. Krugly, “Representation of an invariant measure of irreducible discrete-timemarkov chain with a finite state space by a set of opposite directed trees”, Computer Research and Modeling, 7:2 (2015), 221–226
Linking options:
https://www.mathnet.ru/eng/crm181 https://www.mathnet.ru/eng/crm/v7/i2/p221
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Abstract page: | 120 | Full-text PDF : | 54 | References: | 34 |
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