Characterization by orthogonal polynomial systems of finite Markov chains

2001 ◽  
Vol 38 (A) ◽  
pp. 42-52 ◽  
Author(s):  
E. Seneta
Keyword(s):  
Probability Distribution ◽  
Markov Chains ◽  
Orthogonal Polynomial ◽  
Polynomial System ◽  
Polynomial Systems ◽  
Stochastic Matrices ◽  
Mutation Model ◽  
Hahn Polynomials ◽  
Finite Markov Chains

The paper characterizes matrices which have a given system of vectors orthogonal with respect to a given probability distribution as its right eigenvectors. Results of Hoare and Rahman are unified in this context, then all matrices with a given orthogonal polynomial system as right eigenvectors under the constraint a0j = 0 for j ≥ 2 are specified. The only stochastic matrices P = {pij} satisfying p00 + p01 = 1 with the Hahn polynomials as right eigenvectors have the form of the Moran mutation model.

Characterization by orthogonal polynomial systems of finite Markov chains

2001 ◽  
Vol 38 (A) ◽  
pp. 42-52 ◽  
Author(s):  
E. Seneta
Keyword(s):  
Probability Distribution ◽  
Markov Chains ◽  
Orthogonal Polynomial ◽  
Polynomial System ◽  
Polynomial Systems ◽  
Stochastic Matrices ◽  
Mutation Model ◽  
Hahn Polynomials ◽  
Finite Markov Chains

The paper characterizes matriceswhich have a given system of vectors orthogonal with respect to a given probability distribution as its right eigenvectors. Results of Hoare and Rahman are unified in this context, then all matrices with a given orthogonal polynomial system as right eigenvectors under the constrainta0j= 0 forj≥ 2 are specified. The only stochastic matricesP ={pij} satisfyingp00+p01= 1 with the Hahn polynomials as right eigenvectors have the form of the Moran mutation model.

scholarly journals Semiclassical orthogonal polynomial systems on nonuniform lattices, deformations of the Askey table, and analogues of isomonodromy

2015 ◽  
Vol 219 ◽  
pp. 127-234 ◽  
Author(s):  
N. S. Witte
Keyword(s):  
Orthogonal Polynomial ◽  
Difference Operator ◽  
Divided Difference ◽  
Polynomial System ◽  
Homogeneous Equation ◽  
Polynomial Systems ◽  
Classical Solutions ◽  
First Order ◽  
Wilson Operator ◽  
Spectral Equation

AbstractA 𝔻-semiclassical weight is one which satisfies a particular linear, first-order homogeneous equation in a divided-difference operator 𝔻. It is known that the system of polynomials, orthogonal with respect to this weight, and the associated functions satisfy a linear, first-order homogeneous matrix equation in the divided-difference operator termed thespectral equation. Attached to the spectral equation is a structure which constitutes a number of relations such as those arising from compatibility with the three-term recurrence relation. Here this structure is elucidated in the general case of quadratic lattices. The simplest examples of the 𝔻-semiclassical orthogonal polynomial systems are precisely those in the Askey table of hypergeometric and basic hypergeometric orthogonal polynomials. However within the 𝔻-semiclassical class it is entirely natural to define a generalization of the Askey table weights which involve a deformation with respect to new deformation variables. We completely construct the analogous structures arising from such deformations and their relations with the other elements of the theory. As an example we treat the first nontrivial deformation of the Askey–Wilson orthogonal polynomial system defined by the q-quadratic divided-difference operator, the Askey–Wilson operator, and derive the coupled first-order divided-difference equations characterizing its evolution in the deformation variable. We show that this system is a member of a sequence of classical solutions to theq-Painlevé system.

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.

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 Semiclassical orthogonal polynomial systems on nonuniform lattices, deformations of the Askey table, and analogues of isomonodromy

2015 ◽  
Vol 219 ◽  
pp. 127-234 ◽  
Author(s):  
N. S. Witte
Keyword(s):  
Orthogonal Polynomial ◽  
Difference Operator ◽  
Divided Difference ◽  
Polynomial System ◽  
Homogeneous Equation ◽  
Polynomial Systems ◽  
Classical Solutions ◽  
First Order ◽  
Wilson Operator ◽  
Spectral Equation

AbstractA 𝔻-semiclassical weight is one which satisfies a particular linear, first-order homogeneous equation in a divided-difference operator 𝔻. It is known that the system of polynomials, orthogonal with respect to this weight, and the associated functions satisfy a linear, first-order homogeneous matrix equation in the divided-difference operator termed the spectral equation. Attached to the spectral equation is a structure which constitutes a number of relations such as those arising from compatibility with the three-term recurrence relation. Here this structure is elucidated in the general case of quadratic lattices. The simplest examples of the 𝔻-semiclassical orthogonal polynomial systems are precisely those in the Askey table of hypergeometric and basic hypergeometric orthogonal polynomials. However within the 𝔻-semiclassical class it is entirely natural to define a generalization of the Askey table weights which involve a deformation with respect to new deformation variables. We completely construct the analogous structures arising from such deformations and their relations with the other elements of the theory. As an example we treat the first nontrivial deformation of the Askey–Wilson orthogonal polynomial system defined by the q-quadratic divided-difference operator, the Askey–Wilson operator, and derive the coupled first-order divided-difference equations characterizing its evolution in the deformation variable. We show that this system is a member of a sequence of classical solutions to the q-Painlevé system.

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.

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