scholarly journals Analytic-geometric methods for finite Markov chains with applications to quasi-stationarity

2020 ◽  
Vol 17 (2) ◽  
pp. 901
Author(s):  
Persi Diaconis ◽  
Kelsey Houston-Edwards ◽  
Laurent Saloff-Coste
Keyword(s):  
Markov Chains ◽  
Geometric Methods ◽  
Finite Markov Chains

On quasi-stationary distributions in absorbing continuous-time finite Markov chains

1967 ◽  
Vol 4 (1) ◽  
pp. 192-196 ◽  
Author(s):  
J. N. Darroch ◽  
E. Seneta
Keyword(s):  
Markov Chains ◽  
Discrete Time ◽  
Continuous Time ◽  
Present Note ◽  
Time Parameter ◽  
Stationary Distributions ◽  
Finite State ◽  
Absorbing Markov Chains ◽  
Finite Markov Chains ◽  
Finite State Space

In a recent paper, the authors have discussed the concept of quasi-stationary distributions for absorbing Markov chains having a finite state space, with the further restriction of discrete time. The purpose of the present note is to summarize the analogous results when the time parameter is continuous.

scholarly journals Intersection Conductance and Canonical Alternating Paths: Methods for General Finite Markov Chains

2014 ◽  
Vol 23 (4) ◽  
pp. 585-606
Author(s):  
RAVI MONTENEGRO
Keyword(s):  
Markov Chains ◽  
Cayley Graph ◽  
Mixing Time ◽  
Symmetric Case ◽  
Generating Set ◽  
Finite Markov Chains

We extend the conductance and canonical paths methods to the setting of general finite Markov chains, including non-reversible non-lazy walks. The new path method is used to show that a known bound for the mixing time of a lazy walk on a Cayley graph with a symmetric generating set also applies to the non-lazy non-symmetric case, often even when there is no holding probability.

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 The Ruin Problem for Finite Markov Chains

1991 ◽  
Vol 19 (3) ◽  
pp. 1298-1310 ◽  
Author(s):  
Thomas Hoglund
Keyword(s):  
Markov Chains ◽  
Finite Markov Chains ◽  
Ruin Problem

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.

scholarly journals On sojourn times at particular gene frequencies

1975 ◽  
Vol 25 (2) ◽  
pp. 89-94 ◽  
Author(s):  
Edward Pollak ◽  
Barry C. Arnold
Keyword(s):  
Markov Chains ◽  
Gene Frequency ◽  
Overlapping Generations ◽  
Finite Population ◽  
Diffusion Method ◽  
Gene Frequencies ◽  
Sojourn Times ◽  
The Mean ◽  
Number Of Visits ◽  
Finite Markov Chains

SUMMARYThe distribution of visits to a particular gene frequency in a finite population of size N with non-overlapping generations is derived. It is shown, by using well-known results from the theory of finite Markov chains, that all such distributions are geometric, with parameters dependent only on the set of bij's, where bij is the mean number of visits to frequency j/2N, given initial frequency i/2N. The variance of such a distribution does not agree with the value suggested by the diffusion method. An improved approximation is derived.

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