Approximating the Stationary Distribution of an Infinite Stochastic Matrix

2021 ◽  
pp. 653-654
Author(s):  
Daniel P. Heyman
Keyword(s):  
Stationary Distribution ◽  
Stochastic Matrix

Approximating the stationary distribution of an infinite stochastic matrix

1991 ◽  
Vol 28 (1) ◽  
pp. 96-103 ◽  
Author(s):  
Daniel P. Heyman
Keyword(s):  
Markov Chain ◽  
Steady State ◽  
Stationary Distribution ◽  
Numerical Approximation ◽  
Stochastic Matrix ◽  
Sufficient Conditions ◽  
Finite System ◽  
Balance Equations ◽  
The Given ◽  
Steady State Balance

We are given a Markov chain with states 0, 1, 2, ···. We want to get a numerical approximation of the steady-state balance equations. To do this, we truncate the chain, keeping the first n states, make the resulting matrix stochastic in some convenient way, and solve the finite system. The purpose of this paper is to provide some sufficient conditions that imply that as n tends to infinity, the stationary distributions of the truncated chains converge to the stationary distribution of the given chain. Our approach is completely probabilistic, and our conditions are given in probabilistic terms. We illustrate how to verify these conditions with five examples.

scholarly journals Perturbation of the stationary distribution measured by ergodicity coefficients

1988 ◽  
Vol 20 (01) ◽  
pp. 228-230 ◽  
Author(s):  
E. Seneta
Keyword(s):  
Stationary Distribution ◽  
Stochastic Matrix ◽  
Ergodicity Coefficients

It is shown that an easily calculated ergodicity coefficient of a stochastic matrix P with a unique stationary distribution π T, may be used to assess sensitivity of π T to perturbation of P.

Computation of the stationary distribution of an infinite stochastic matrix of special form

1974 ◽  
Vol 10 (2) ◽  
pp. 255-261 ◽  
Author(s):  
G.H. Golub ◽  
E. Seneta
Keyword(s):  
Stationary Distribution ◽  
Special Form ◽  
Stochastic Matrix ◽  
Structural Form

An algorithm is presented for computing the unique stationary distribution of an infinite regular stochastic matrix of a structural form subsuming both upper-Hessenberg and generalized renewal matrices of this kind. Convergence is elementwise, monotone from above, from information within finite truncations, of increasing order.

Approximating the stationary distribution of an infinite stochastic matrix

1991 ◽  
Vol 28 (01) ◽  
pp. 96-103 ◽  
Author(s):  
Daniel P. Heyman
Keyword(s):  
Markov Chain ◽  
Steady State ◽  
Stationary Distribution ◽  
Numerical Approximation ◽  
Stochastic Matrix ◽  
Sufficient Conditions ◽  
Finite System ◽  
Balance Equations ◽  
The Given ◽  
Steady State Balance

We are given a Markov chain with states 0, 1, 2, ···. We want to get a numerical approximation of the steady-state balance equations. To do this, we truncate the chain, keeping the first n states, make the resulting matrix stochastic in some convenient way, and solve the finite system. The purpose of this paper is to provide some sufficient conditions that imply that as n tends to infinity, the stationary distributions of the truncated chains converge to the stationary distribution of the given chain. Our approach is completely probabilistic, and our conditions are given in probabilistic terms. We illustrate how to verify these conditions with five examples.

scholarly journals Computation of the stationary distribution of an infinite Markov matrix

1973 ◽  
Vol 8 (3) ◽  
pp. 333-341 ◽  
Author(s):  
G.H. Golub ◽  
E. Seneta
Keyword(s):  
Convergence Rate ◽  
Stationary Distribution ◽  
Stochastic Matrix ◽  
Markov Matrix

An algorithm is presented for computing the unique stationary distribution of an infinite stochastic matrix possessing at least one column whose elements are bounded away from zero. Elementwise convergence rate is discussed by means of two examples.

scholarly journals A Numerical Algorithm on the Computation of the Stationary Distribution of a Discrete Time Homogenous Finite Markov Chain

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Di Zhao ◽  
Hongyi Li ◽  
Donglin Su
Keyword(s):  
Markov Chain ◽  
Stationary Distribution ◽  
Discrete Time ◽  
Transition Matrix ◽  
Stochastic Matrix ◽  
Nonnegative Matrix ◽  
Homogeneous Markov Chain ◽  
Irreducible Matrix ◽  
Nonnegative Irreducible Matrix ◽  
Perron Vector

The transition matrix, which characterizes a discrete time homogeneous Markov chain, is a stochastic matrix. A stochastic matrix is a special nonnegative matrix with each row summing up to 1. In this paper, we focus on the computation of the stationary distribution of a transition matrix from the viewpoint of the Perron vector of a nonnegative matrix, based on which an algorithm for the stationary distribution is proposed. The algorithm can also be used to compute the Perron root and the corresponding Perron vector of any nonnegative irreducible matrix. Furthermore, a numerical example is given to demonstrate the validity of the algorithm.

scholarly journals Perturbation of the stationary distribution measured by ergodicity coefficients

1988 ◽  
Vol 20 (1) ◽  
pp. 228-230 ◽  
Author(s):  
E. Seneta
Keyword(s):  
Stationary Distribution ◽  
Stochastic Matrix ◽  
Ergodicity Coefficients

It is shown that an easily calculated ergodicity coefficient of a stochastic matrix P with a unique stationary distribution πT, may be used to assess sensitivity of πT to perturbation of P.

Normal approximations to discrete unimodal distributions

1986 ◽  
Vol 23 (04) ◽  
pp. 1013-1018
Author(s):  
B. G. Quinn ◽  
H. L. MacGillivray
Keyword(s):  
Stationary Distribution ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Hypergeometric Distribution ◽  
Death Process ◽  
Normal Approximations ◽  
Discrete Random Variables ◽  
Birth Death

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

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