scholarly journals Several Types of Ergodicity for M/G/1-Type Markov Chains and Markov Processes

2006 ◽  
Vol 43 (01) ◽  
pp. 141-158 ◽  
Author(s):  
Yuanyuan Liu ◽  
Zhenting Hou
Keyword(s):  
Markov Chains ◽  
Markov Processes ◽  
Generating Function ◽  
Final Section ◽  
Convergence Rates ◽  
Exponential Convergence ◽  
Radius Of Convergence ◽  
Exponential Ergodicity ◽  
Return Probability ◽  
Explicit Bounds

In this paper we study polynomial and geometric (exponential) ergodicity for M/G/1-type Markov chains and Markov processes. First, practical criteria for M/G/1-type Markov chains are obtained by analyzing the generating function of the first return probability to level 0. Then the corresponding criteria for M/G/1-type Markov processes are given, using their h-approximation chains. Our method yields the radius of convergence of the generating function of the first return probability, which is very important in obtaining explicit bounds on geometric (exponential) convergence rates. Our results are illustrated, in the final section, in some examples.

scholarly journals Several Types of Ergodicity for M/G/1-Type Markov Chains and Markov Processes

2006 ◽  
Vol 43 (1) ◽  
pp. 141-158 ◽  
Author(s):  
Yuanyuan Liu ◽  
Zhenting Hou
Keyword(s):  
Markov Chains ◽  
Markov Processes ◽  
Generating Function ◽  
Final Section ◽  
Convergence Rates ◽  
Exponential Convergence ◽  
Radius Of Convergence ◽  
Exponential Ergodicity ◽  
Return Probability ◽  
Explicit Bounds

In this paper we study polynomial and geometric (exponential) ergodicity for M/G/1-type Markov chains and Markov processes. First, practical criteria for M/G/1-type Markov chains are obtained by analyzing the generating function of the first return probability to level 0. Then the corresponding criteria for M/G/1-type Markov processes are given, using their h-approximation chains. Our method yields the radius of convergence of the generating function of the first return probability, which is very important in obtaining explicit bounds on geometric (exponential) convergence rates. Our results are illustrated, in the final section, in some examples.

Explicit bounds for geometric convergence of Markov chains

2000 ◽  
Vol 37 (3) ◽  
pp. 642-651
Author(s):  
John E. Kolassa
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Generalized Linear Models ◽  
Linear Models ◽  
Convergence Rates ◽  
Conditional Inference ◽  
Transition Kernel ◽  
Geometric Convergence ◽  
Explicit Bounds ◽  
Transition Operators

This paper presents bounds on convergence rates of Markov chains in terms of quantities calculable directly from chain transition operators. Bounds are constructed by creating a probability distribution that minorizes the transition kernel over some region, and by examining bounds on an expectation conditional on lying within and without this region. These are shown to be sharper in most cases than previous similar results. These bounds are applied to a Markov chain useful in frequentist conditional inference in canonical generalized linear models.

scholarly journals Computable exponential convergence rates for stochastically ordered Markov processes

1996 ◽  
Vol 6 (1) ◽  
pp. 218-237 ◽  
Author(s):  
Robert B. Lund ◽  
Sean P. Meyn ◽  
Richard L. Tweedie
Keyword(s):  
Markov Processes ◽  
Convergence Rates ◽  
Exponential Convergence

Explicit bounds for geometric convergence of Markov chains

2000 ◽  
Vol 37 (03) ◽  
pp. 642-651 ◽  
Author(s):  
John E. Kolassa
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Generalized Linear Models ◽  
Linear Models ◽  
Convergence Rates ◽  
Conditional Inference ◽  
Transition Kernel ◽  
Geometric Convergence ◽  
Explicit Bounds ◽  
Transition Operators

This paper presents bounds on convergence rates of Markov chains in terms of quantities calculable directly from chain transition operators. Bounds are constructed by creating a probability distribution that minorizes the transition kernel over some region, and by examining bounds on an expectation conditional on lying within and without this region. These are shown to be sharper in most cases than previous similar results. These bounds are applied to a Markov chain useful in frequentist conditional inference in canonical generalized linear models.

On the absorption probabilities and mean time for absorption for discrete Markov chains

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nikolaos Halidias
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Generating Function ◽  
Discrete Time ◽  
Probability Generating Function ◽  
The Mean ◽  
Mean Time ◽  
Absorption Probabilities

Abstract In this note we study the probability and the mean time for absorption for discrete time Markov chains. In particular, we are interested in estimating the mean time for absorption when absorption is not certain and connect it with some other known results. Computing a suitable probability generating function, we are able to estimate the mean time for absorption when absorption is not certain giving some applications concerning the random walk. Furthermore, we investigate the probability for a Markov chain to reach a set A before reach B generalizing this result for a sequence of sets A 1 , A 2 , … , A k {A_{1},A_{2},\dots,A_{k}} .

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