Exponential Convergence Rates of Markov Chains under a Weaken Minorization Condition

2018 ◽  
Vol 34 (12) ◽  
pp. 1829-1836
Author(s):  
Shu Lan Hu ◽  
Xin Yu Wang
Keyword(s):  
Markov Chains ◽  
Convergence Rates ◽  
Exponential Convergence ◽  
Minorization Condition

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.

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.

Stability and exponential convergence of continuous-time Markov chains

2003 ◽  
Vol 40 (04) ◽  
pp. 970-979 ◽  
Author(s):  
A. Yu. Mitrophanov
Keyword(s):  
Markov Chains ◽  
Rate Of Convergence ◽  
Stationary Distribution ◽  
Continuous Time ◽  
Exponential Convergence ◽  
Perturbation Bounds ◽  
Continuous Time Markov Chains ◽  
The Relationship

For finite, homogeneous, continuous-time Markov chains having a unique stationary distribution, we derive perturbation bounds which demonstrate the connection between the sensitivity to perturbations and the rate of exponential convergence to stationarity. Our perturbation bounds substantially improve upon the known results. We also discuss convergence bounds for chains with diagonalizable generators and investigate the relationship between the rate of convergence and the sensitivity of the eigenvalues of the generator; special attention is given to reversible chains.

scholarly journals EXPONENTIAL CONVERGENCE OF hp-FEM FOR MAXWELL EQUATIONS WITH WEIGHTED REGULARIZATION IN POLYGONAL DOMAINS

2005 ◽  
Vol 15 (04) ◽  
pp. 575-622 ◽  
Author(s):  
MARTIN COSTABEL ◽  
MONIQUE DAUGE ◽  
CHRISTOPH SCHWAB
Keyword(s):  
Finite Element ◽  
Maxwell’S Equations ◽  
Maxwell Equations ◽  
Maxwell's Equations ◽  
Convergence Rates ◽  
Exponential Convergence ◽  
Polygonal Domains ◽  
Time Harmonic ◽  
Standard Tool ◽  
Optimal Convergence Rates

The time-harmonic Maxwell equations do not have an elliptic nature by themselves. Their regularization by a divergence term is a standard tool to obtain equivalent elliptic problems. Nodal finite element discretizations of Maxwell's equations obtained from such a regularization converge to wrong solutions in any non-convex polygon. Modification of the regularization term consisting in the introduction of a weight restores the convergence of nodal FEM, providing optimal convergence rates for the h version of finite elements. We prove exponential convergence of hp FEM for the weighted regularization of Maxwell's equations in plane polygonal domains provided the hp-FE spaces satisfy a series of axioms. We verify these axioms for several specific families of hp finite element spaces.

On records and related processes for sequences with trends

1999 ◽  
Vol 36 (3) ◽  
pp. 668-681 ◽  
Author(s):  
K. Borovkov
Keyword(s):  
Markov Chains ◽  
Limit Theorems ◽  
Convergence Rates ◽  
Rate Function ◽  
Regular Case ◽  
Limiting Distributions ◽  
Record Times ◽  
Ergodicity Theorem ◽  
Linear Trends

We study the records and related variables for sequences with linear trends. We discuss the properties of the asymptotic rate function and relationships between the distribution of the long-term maxima in the sequence and that of a particular observation, including two characterization type results. We also consider certain Markov chains related to the process of records and prove limit theorems for them, including the ergodicity theorem in the regular case (convergence rates are given under additional assumptions), and derive the limiting distributions for the inter-record times and increments of records.

Exponential convergence rates for weighted sums in noncommutative probability space

2016 ◽  
Vol 19 (04) ◽  
pp. 1650027 ◽  
Author(s):  
Byoung Jin Choi ◽  
Un Cig Ji
Keyword(s):  
Probability Space ◽  
Von Neumann Algebra ◽  
Convergence Rates ◽  
Exponential Convergence ◽  
Large Deviation ◽  
Type Inequality ◽  
Independent Random Variables ◽  
Weighted Sums ◽  
Noncommutative Probability ◽  
Von Neumann

We study exponential convergence rates for weighted sums of successive independent random variables in a noncommutative probability space of which the weights are in a von Neumann algebra. Then we prove a noncommutative extension of the result for the exponential convergence rate by Baum, Katz and Read. As applications, we first study a large deviation type inequality for weighted sums in a noncommutative probability space, and secondly we study exponential convergence rates for weighted free additive convolution sums of probability measures.

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