## Continuous-Time Markov Chains : An Applications-Oriented Approach

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SearchWorks Catalog Stanford Libraries. Responsibility by William J. Physical description 1 online resource xii, pages 5 illustrations.

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Series Springer series in statistics. Probability and its applications. Online Available online. SpringerLink Full view. More options. Find it at other libraries via WorldCat Limited preview. Contents 1 Transition Functions and Resolvents. Resolvent Functions and Their Properties. Feller Transition Functions.

Kendall's Representation of Reversible Transition Functions. Existence and Uniqueness of Q-Functions. Finite Markov Chains. Birth and Death Processes. Laplace Transform Tools.

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## Continuous Time Markov Chains - An Applications Oriented Approach (Hardcover)

Uniqueness-The Non-Conservative Case. Classification of States. Sub-invariant and Invariant Measures.

Classification Based on the Q-Matrix. Determination of Invariant Measures from the Q-Matrix.

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The Ergodic Coefficient and Hitting Times. Ordinary Ergodicity. However, their methods are not fitted for the general continuous time Markov chains, especially when the symmetric condition, coupling condition or stochastically monotone one is not satisfied. For example, the bounds of Markov chains with instantaneous states such as Kolmogorov matrix, or the regular birth and death process.

## Probability Surveys - Vol. 7 ()

In this paper, we discuss this problem. Moreover , if. In this paper we shall first develop the methods in [ 2 ] to the continuous time situation, which leads to considerable improvements of convergence rates. And this result shall be in a wider range of application than existing results in [ 5 — 7 ]. Next we shall give some fundamental lemmas and the proof of the main theorem in this paper.

From 1 together with Theorems 2. Then the stopping times mentioned above are almost surely finite and. By 9 and the equations above we have. And by 8 we have. By 11 we have. By 3 and 4 we obtain.

### ISBN 10: 1461277728

By 12 we get. Then we know that. In this section we compute the maximal exponentially ergodic constants for two types of chains: a kind of singular Markov chain in which all states are not conservative and Kolmogorov matrix in which state 1 is an instantaneous state. From [ 5 ] it is known that this chain is not symmetric, so we cannot discuss its ergodicity with coupling theory. We also cannot adapt existing results to this chain.

The following are our main methods and result. Then by 15 we have. In the following we discuss the conditions of exponential ergodicity and convergence rate of exponential ergodicity. This matrix is called the Kolmogorov matrix. There are infinitely many dishonest processes with this Q -matrix.

The authors see [ 8 , 9 , 14 ] have shown that the process with the following resolvents is the only honest one. Though this chain is weakly symmetric, its convergence rate is still unknown because of its instantaneous state. In the following we discuss the conditions of exponential ergodicity and the convergence rates of exponential ergodicity. According to the results above, the maximal exponentially ergodic constant of this example satisfies.

Baxendale, PH: Renewal theory and computable convergence rates for geometrically ergodic Markov chains. Gani and E. Springer, New York Hunan Science and Technology Press, Changsha Wiley, Chichester