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.

Perturbation theory and finite Markov chains

1968 ◽  
Vol 5 (02) ◽  
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 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.

General transition probabilities for finite Markov chains

1964 ◽  
Vol 60 (1) ◽  
pp. 83-91 ◽  
Author(s):  
Marcel F. Neuts
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Finite Number ◽  
Discrete Time ◽  
Transition Probabilities ◽  
Initial State ◽  
Discrete Time Markov Chain ◽  
Finite Markov Chains

We consider a stationary discrete-time Markov chain with a finite number m of possible states which we designate by 1,…,m. We assume that at time t = 0 the process is in an initial state i with probability (i = 1,…, m) and such that and .

On Quasi-Stationary distributions in absorbing discrete-time finite Markov chains

1965 ◽  
Vol 2 (1) ◽  
pp. 88-100 ◽  
Author(s):  
J. N. Darroch ◽  
E. Seneta
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Markov Chains ◽  
Stationary Distribution ◽  
Discrete Time ◽  
Stationary Distributions ◽  
Transient States ◽  
Reverse Process ◽  
Finite Markov Chains ◽  
Quasi Stationary Distribution

The time to absorption from the set T of transient states of a Markov chain may be sufficiently long for the probability distribution over T to settle down in some sense to a “quasi-stationary” distribution. Various analogues of the stationary distribution of an irreducible chain are suggested and compared. The reverse process of an absorbing chain is found to be relevant.

XXIV.—Conditions for the Ergodicity of Non-homogeneous Finite Markov Chains

1957 ◽  
Vol 64 (4) ◽  
pp. 369-380 ◽  
Author(s):  
J. L. Mott
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Asymptotic Behaviour ◽  
Transition Probabilities ◽  
Finite Markov Chain ◽  
Finite Markov Chains ◽  
Product Of Matrices

SynopsisIn this note we study the asymptotic behaviour of a product of matrices where Pj is a matrix of transition probabilities in a non-homogeneous finite Markov chain. We give conditions that (i) the rows of P(n) tend to identity and that (ii) P(n) tends to a limit matrix with identical rows.

On Quasi-Stationary distributions in absorbing discrete-time finite Markov chains

1965 ◽  
Vol 2 (01) ◽  
pp. 88-100 ◽  
Author(s):  
J. N. Darroch ◽  
E. Seneta
Keyword(s):  
Markov Chain ◽  
Probability Distribution ◽  
Markov Chains ◽  
Stationary Distribution ◽  
Discrete Time ◽  
Stationary Distributions ◽  
Transient States ◽  
Reverse Process ◽  
Finite Markov Chains ◽  
Quasi Stationary Distribution

The time to absorption from the set T of transient states of a Markov chain may be sufficiently long for the probability distribution over T to settle down in some sense to a “quasi-stationary” distribution. Various analogues of the stationary distribution of an irreducible chain are suggested and compared. The reverse process of an absorbing chain is found to be relevant.

scholarly journals A Bayesian model for binary Markov chains

2004 ◽  
Vol 2004 (8) ◽  
pp. 421-429 ◽  
Author(s):  
Souad Assoudou ◽  
Belkheir Essebbar
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chains ◽  
Bayesian Estimation ◽  
Bayesian Model ◽  
Transition Probabilities ◽  
Simulated Data ◽  
Bayesian Estimator ◽  
Jeffreys Prior ◽  
Data Set

This note is concerned with Bayesian estimation of the transition probabilities of a binary Markov chain observed from heterogeneous individuals. The model is founded on the Jeffreys' prior which allows for transition probabilities to be correlated. The Bayesian estimator is approximated by means of Monte Carlo Markov chain (MCMC) techniques. The performance of the Bayesian estimates is illustrated by analyzing a small simulated data set.

scholarly journals Infinitely divisible random transition probabilities with application to dependent Markov chains

1984 ◽  
Vol 25 (4) ◽  
pp. 463-472
Author(s):  
Peter L. Chesson
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
White Noise ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Transition Rates ◽  
Infinitely Divisible ◽  
Random Transition ◽  
Transition Probability Matrices ◽  
Probability Matrices

AbstractRandom transition probability matrices with stationary independent factors define “white noise” environment processes for Markov chains. Two examples are considered in detail. Such environment processes can be used to construct several Markov chains which are dependent, have the same transition probabilities and are jointly a Markov chain. Transition rates for such processes are evaluated. These results have application to the study of animal movements.

Exact and ordinary lumpability in finite Markov chains

1994 ◽  
Vol 31 (1) ◽  
pp. 59-75 ◽  
Author(s):  
Peter Buchholz
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Exact Determination ◽  
Finite Markov Chains

Exact and ordinary lumpability in finite Markov chains is considered. Both concepts naturally define an aggregation of the Markov chain yielding an aggregated chain that allows the exact determination of several stationary and transient results for the original chain. We show which quantities can be determined without an error from the aggregated process and describe methods to calculate bounds on the remaining results. Furthermore, the concept of lumpability is extended to near lumpability yielding approximative aggregation.

Single-shelf library-type Markov chains with infinitely many books

1977 ◽  
Vol 14 (02) ◽  
pp. 298-308 ◽  
Author(s):  
Peter R. Nelson
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Transition Probabilities ◽  
Natural Order ◽  
Order Book ◽  
Exit Boundary ◽  
One Step ◽  
Step Transition ◽  
The Right ◽  
Positive Recurrent

In a single-shelf library having infinitely many books B 1 , B 2 , …, the probability of selecting each book is assumed known. Books are removed one at a time and replaced in position k prior to the next removal. Books are moved either to the right or the left as is necessary to vacate position k. Those arrangements of books where after some finite position all the books are in natural order (book i occupies position i) are considered as states in an infinite Markov chain. When k > 1, we show that the chain can never be positive recurrent. When k = 1, we find the limits of ratios of one-step transition probabilities; and when k = 1 and the chain is transient, we find the Martin exit boundary.

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