Spectral representations of the transition probability matrices for continuous time finite Markov chains

1996 ◽  
Vol 33 (1) ◽  
pp. 28-33 ◽  
Author(s):  
Nan Fu Peng
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Transition Probability ◽  
Algebraic Method ◽  
Finite State ◽  
Transition Probability Matrices ◽  
Finite Markov Chains ◽  
Probability Matrices ◽  
Spectral Representations

Using an easy linear-algebraic method, we obtain spectral representations, without the need for eigenvector determination, of the transition probability matrices for completely general continuous time Markov chains with finite state space. Comparing the proof presented here with that of Brown (1991), who provided a similar result for a special class of finite Markov chains, we observe that ours is more concise.

Spectral representations of the transition probability matrices for continuous time finite Markov chains

1996 ◽  
Vol 33 (01) ◽  
pp. 28-33 ◽  
Author(s):  
Nan Fu Peng
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Transition Probability ◽  
Algebraic Method ◽  
Finite State ◽  
Transition Probability Matrices ◽  
Finite Markov Chains ◽  
Probability Matrices ◽  
Spectral Representations

Using an easy linear-algebraic method, we obtain spectral representations, without the need for eigenvector determination, of the transition probability matrices for completely general continuous time Markov chains with finite state space. Comparing the proof presented here with that of Brown (1991), who provided a similar result for a special class of finite Markov chains, we observe that ours is more concise.

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 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.

A multinomial model for transition probability matrices

1975 ◽  
Vol 12 (03) ◽  
pp. 498-506 ◽  
Author(s):  
G. G. S. Pegram
Keyword(s):  
Markov Chains ◽  
Conventional Method ◽  
Transition Probability ◽  
Multinomial Model ◽  
Arbitrary Size ◽  
Current Estimation ◽  
Transition Probability Matrices ◽  
Probability Matrices ◽  
Two Parameters

A model for the transition probability matrices (t.p.m.'s) of finite discrete Markov chains is suggested which may help those who wish to use a larger number of states than would seem reasonable with the data available in the current estimation situation. The model is especially useful in that a finite t.p.m. of arbitrary size can be specified by as few as two parameters. An example of the model's estimation and use is presented, showing it in a fair light in comparison with the conventional method of t.p.m. specification.

A multinomial model for transition probability matrices

1975 ◽  
Vol 12 (3) ◽  
pp. 498-506 ◽  
Author(s):  
G. G. S. Pegram
Keyword(s):  
Markov Chains ◽  
Conventional Method ◽  
Transition Probability ◽  
Multinomial Model ◽  
Arbitrary Size ◽  
Current Estimation ◽  
Transition Probability Matrices ◽  
Probability Matrices ◽  
Two Parameters

A model for the transition probability matrices (t.p.m.'s) of finite discrete Markov chains is suggested which may help those who wish to use a larger number of states than would seem reasonable with the data available in the current estimation situation. The model is especially useful in that a finite t.p.m. of arbitrary size can be specified by as few as two parameters. An example of the model's estimation and use is presented, showing it in a fair light in comparison with the conventional method of t.p.m. specification.

scholarly journals Why is Kemeny’s constant a constant?

2018 ◽  
Vol 55 (4) ◽  
pp. 1025-1036 ◽  
Author(s):  
Dario Bini ◽  
Jeffrey J. Hunter ◽  
Guy Latouche ◽  
Beatrice Meini ◽  
Peter Taylor
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Continuous Time ◽  
Death Process ◽  
Birth And Death Process ◽  
Finite State ◽  
Finite Markov Chains ◽  
Infinite State ◽  
Special Case ◽  
Finite State Space

Abstract In their 1960 book on finite Markov chains, Kemeny and Snell established that a certain sum is invariant. The value of this sum has become known as Kemeny’s constant. Various proofs have been given over time, some more technical than others. We give here a very simple physical justification, which extends without a hitch to continuous-time Markov chains on a finite state space. For Markov chains with denumerably infinite state space, the constant may be infinite and even if it is finite, there is no guarantee that the physical argument will hold. We show that the physical interpretation does go through for the special case of a birth-and-death process with a finite value of Kemeny’s constant.

CLOSED-FORM STOCHASTIC BOUNDS ON THE STATIONARY DISTRIBUTION OF MARKOV CHAINS

2002 ◽  
Vol 16 (4) ◽  
pp. 403-426 ◽  
Author(s):  
Mouad Ben Mamoun ◽  
Nihal Pekergin
Keyword(s):  
Markov Chains ◽  
Closed Form ◽  
Stationary Distribution ◽  
Discrete Time ◽  
Transition Probability ◽  
Matrix Structure ◽  
Stochastic Monotonicity ◽  
Monotonicity Properties ◽  
Transition Probability Matrices ◽  
Probability Matrices

We propose a particular class of transition probability matrices for discrete-time Markov chains with a closed form to compute the stationary distribution. The stochastic monotonicity properties of this class are established. We give algorithms to construct monotone, bounding matrices belonging to the proposed class for the variability orders. The accuracy of bounds with respect to the underlying matrix structure is discussed through an example.

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

1967 ◽  
Vol 4 (01) ◽  
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.

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.

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