An extension of a theorem concerning an interesting Markov chain

1973 ◽  
Vol 10 (04) ◽  
pp. 886-890 ◽  
Author(s):  
W. J. Hendricks
Keyword(s):  
Markov Chain ◽  
Stationary Distribution ◽  
Ergodic Markov Chain ◽  
The Right

In a single-shelf library of N books we suppose that books are selected one at a time and returned to the kth position on the shelf before another selection is made. Books are moved to the right or left as necessary to vacate position k. The probability of selecting each book is assumed to be known, and the N! arrangements of the books are considered as states of an ergodic Markov chain for which we find the stationary distribution.

An extension of a theorem concerning an interesting Markov chain

1973 ◽  
Vol 10 (4) ◽  
pp. 886-890 ◽  
Author(s):  
W. J. Hendricks
Keyword(s):  
Markov Chain ◽  
Stationary Distribution ◽  
Ergodic Markov Chain ◽  
The Right

In a single-shelf library of N books we suppose that books are selected one at a time and returned to the kth position on the shelf before another selection is made. Books are moved to the right or left as necessary to vacate position k. The probability of selecting each book is assumed to be known, and the N! arrangements of the books are considered as states of an ergodic Markov chain for which we find the stationary distribution.

scholarly journals Normal distributions of finite Markov chains

2019 ◽  
Vol 29 (08) ◽  
pp. 1431-1449
Author(s):  
John Rhodes ◽  
Anne Schilling
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Stationary Distribution ◽  
Planar Graphs ◽  
Finite Semigroup ◽  
Finite Markov Chain ◽  
Normal Distributions ◽  
Straight Line ◽  
Finite Markov Chains ◽  
The Right

We show that the stationary distribution of a finite Markov chain can be expressed as the sum of certain normal distributions. These normal distributions are associated to planar graphs consisting of a straight line with attached loops. The loops touch only at one vertex either of the straight line or of another attached loop. Our analysis is based on our previous work, which derives the stationary distribution of a finite Markov chain using semaphore codes on the Karnofsky–Rhodes and McCammond expansion of the right Cayley graph of the finite semigroup underlying the Markov chain.

The stationary distribution of an interesting Markov chain

1972 ◽  
Vol 9 (1) ◽  
pp. 231-233 ◽  
Author(s):  
W. J. Hendricks
Keyword(s):  
Markov Chain ◽  
Stationary Distribution ◽  
Ergodic Markov Chain

In a single-shelf library of N books we suppose that books are returned by placing them at one end of the shelf. The probability of selecting each book is assumed to be known, and only one book is removed at a time. The N! arrangements of the books are considered as states of an ergodic Markov chain for which we find the stationary distribution.

The stationary distribution of an interesting Markov chain

1972 ◽  
Vol 9 (01) ◽  
pp. 231-233 ◽  
Author(s):  
W. J. Hendricks
Keyword(s):  
Markov Chain ◽  
Stationary Distribution ◽  
Ergodic Markov Chain

In a single-shelf library of N books we suppose that books are returned by placing them at one end of the shelf. The probability of selecting each book is assumed to be known, and only one book is removed at a time. The N! arrangements of the books are considered as states of an ergodic Markov chain for which we find the stationary distribution.

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 Stationary distribution Markov chain for Covid-19 pandemic

2021 ◽  
Vol 1722 ◽  
pp. 012084
Author(s):  
A L H Achmad ◽  
Mahrudinda ◽  
B N Ruchjana
Keyword(s):  
Markov Chain ◽  
Stationary Distribution

Perturbation theory and finite Markov chains

1968 ◽  
Vol 5 (2) ◽  
pp. 401-413 ◽  
Author(s):  
Paul J. Schweitzer
Keyword(s):  
Perturbation Theory ◽  
Markov Chain ◽  
Markov Chains ◽  
Stationary Distribution ◽  
Fundamental Matrix ◽  
Transition Probabilities ◽  
Semi Group ◽  
Partial Derivatives ◽  
Finite Markov Chains ◽  
Derivatives Of

A perturbation formalism is presented which shows how the stationary distribution and fundamental matrix of a Markov chain containing a single irreducible set of states change as the transition probabilities vary. Expressions are given for the partial derivatives of the stationary distribution and fundamental matrix with respect to the transition probabilities. Semi-group properties of the generators of transformations from one Markov chain to another are investigated. It is shown that a perturbation formalism exists in the multiple subchain case if and only if the change in the transition probabilities does not alter the number of, or intermix the various subchains. The formalism is presented when this condition is satisfied.

scholarly journals A Combinatorial Derivation of the PASEP Stationary State

10.37236/1134 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Richard Brak ◽  
Sylvie Corteel ◽  
John Essam ◽  
Robert Parviainen ◽  
Andrew Rechnitzer
Keyword(s):  
Markov Chain ◽  
Stationary State ◽  
Stationary Distribution ◽  
Exclusion Process ◽  
Direct Proof ◽  
Asymmetric Exclusion Process ◽  
Original Process ◽  
Combinatorial Derivation ◽  
Path Models ◽  
Asymmetric Exclusion

We give a combinatorial derivation and interpretation of the weights associated with the stationary distribution of the partially asymmetric exclusion process. We define a set of weight equations, which the stationary distribution satisfies. These allow us to find explicit expressions for the stationary distribution and normalisation using both recurrences and path models. To show that the stationary distribution satisfies the weight equations, we construct a Markov chain on a larger set of generalised configurations. A bijection on this new set of configurations allows us to find the stationary distribution of the new chain. We then show that a subset of the generalised configurations is equivalent to the original process and that the stationary distribution on this subset is simply related to that of the original chain. We also provide a direct proof of the validity of the weight equations.

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