scholarly journals Stability Estimates for Finite-Dimensional Distributions of Time-Inhomogeneous Markov Chains

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 174
Author(s):  
Vitaliy Golomoziy ◽  
Yuliya Mishura
Keyword(s):  
Markov Chains ◽  
Transition Probabilities ◽  
Absolute Difference ◽  
Stability Estimates ◽  
Finite Dimensional ◽  
Finite Dimensional Distribution ◽  
Inhomogeneous Markov Chain ◽  
Dimensional Distribution ◽  
General State Space ◽  
The Stability

This paper is devoted to the study of the stability of finite-dimensional distribution of time-inhomogeneous, discrete-time Markov chains on a general state space. The main result of the paper provides an estimate for the absolute difference of finite-dimensional distributions of a given time-inhomogeneous Markov chain and its perturbed version. By perturbation, we mean here small changes in the transition probabilities. Stability estimates are obtained using the coupling method.

Stability of functionals of perturbed Markov chains under the condition of uniform minorization

2020 ◽  
Vol 28 (4) ◽  
pp. 237-251
Author(s):  
Vitaliy Golomoziy
Keyword(s):  
Markov Chains ◽  
Large Deviations ◽  
State Space ◽  
Discrete Time ◽  
Stability Estimates ◽  
General State ◽  
Whole Space ◽  
Minorization Condition ◽  
General State Space ◽  
The Stability

AbstractIn this paper, we investigate the stability of functionals and trajectories of two different, independent, time-inhomogeneous, discrete-time Markov chains on a general state space. We obtain various stability estimates such as an estimate for a difference in expectations of functionals, {L_{2}} stability, and a probability of large deviations. The key condition that is used is the minorization condition on the whole space. We consider different limitations on the functional and on the proximity of two chains. We use the coupling method as a primary technique in our proofs.

The robustness of positive recurrence and recurrence of Markov chains under perturbations of the transition probabilities

1975 ◽  
Vol 12 (04) ◽  
pp. 744-752 ◽  
Author(s):  
Richard L. Tweedie
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Finite Number ◽  
Transition Probabilities ◽  
General State ◽  
Transition Matrices ◽  
Positive Recurrence ◽  
Markov Chain Models ◽  
Countable Space ◽  
General State Space

In many Markov chain models, the immediate characteristic of importance is the positive recurrence of the chain. In this note we investigate whether positivity, and also recurrence, are robust properties of Markov chains when the transition laws are perturbed. The chains we consider are on a fairly general state space : when specialised to a countable space, our results are essentially that, if the transition matrices of two irreducible chains coincide on all but a finite number of columns, then positivity of one implies positivity of both; whilst if they coincide on all but a finite number of rows and columns, recurrence of one implies recurrence of both. Examples are given to show that these results (and their general analogues) cannot in general be strengthened.

The point-process approach to age- and time-dependent branching processes

1983 ◽  
Vol 15 (01) ◽  
pp. 1-20
Author(s):  
Marek Kimmel
Keyword(s):  
Point Process ◽  
Branching Process ◽  
Cell Kinetics ◽  
Branching Processes ◽  
Finite Dimensional ◽  
Finite Dimensional Distribution ◽  
Symbolic Method ◽  
Age Dependent ◽  
Dimensional Distribution ◽  
Varying Environment

The multitype age-dependent branching process in varying environment is treated as a random stream (point process) of the birth and death events. We derive the point-process version of the Bellman–Harris equation, which provides us with a symbolic method of writing the equations for the arbitrary finite-dimensional distribution of the process. We also derive a new recurrence relation, the ‘principle of last generation'. This result is applied to obtain new moment equations for the process with immigration also. General considerations are illustrated by a simple cell kinetics model.

The point-process approach to age- and time-dependent branching processes

1983 ◽  
Vol 15 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Marek Kimmel
Keyword(s):  
Point Process ◽  
Branching Process ◽  
Cell Kinetics ◽  
Branching Processes ◽  
Finite Dimensional ◽  
Finite Dimensional Distribution ◽  
Symbolic Method ◽  
Age Dependent ◽  
Dimensional Distribution ◽  
Varying Environment

The multitype age-dependent branching process in varying environment is treated as a random stream (point process) of the birth and death events. We derive the point-process version of the Bellman–Harris equation, which provides us with a symbolic method of writing the equations for the arbitrary finite-dimensional distribution of the process. We also derive a new recurrence relation, the ‘principle of last generation'. This result is applied to obtain new moment equations for the process with immigration also. General considerations are illustrated by a simple cell kinetics model.

scholarly journals On Weak Limiting Distributions for Random Walks on a Spider

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2000
Author(s):  
Youngsoo Seol
Keyword(s):  
Brownian Motion ◽  
Random Walks ◽  
Classical Case ◽  
Weak Limit ◽  
Skew Brownian Motion ◽  
Finite Dimensional ◽  
Finite Dimensional Distribution ◽  
Dimensional Distribution ◽  
Functional Central Limit ◽  
Special Case

In this article, we study random walks on a spider that can be established from the classical case of simple symmetric random walks. The primary purpose of this article is to establish a functional central limit theorem for random walks on a spider and to define Brownian spider as the resulting weak limit. In special case, random walks on a spider can be characterized as skew random walks. It is well known for skew Brownian motion as the resulting weak limit of skew random walks. We first will study the tightness and then it will be shown for the convergence of finite dimensional distribution for random walks on a spider.

scholarly journals Spectral Theory for Weakly Reversible Markov Chains

2012 ◽  
Vol 49 (01) ◽  
pp. 245-265 ◽  
Author(s):  
Achim Wübker
Keyword(s):  
Markov Chains ◽  
Spectral Gap ◽  
Transition Probabilities ◽  
Markov Operator ◽  
Spectral Gaps ◽  
Reversible Markov Chains ◽  
General State Space ◽  
Stationary Transition ◽  
Positive Recurrent ◽  
Necessary And Sufficient

The theory of L 2-spectral gaps for reversible Markov chains has been studied by many authors. In this paper we consider positive recurrent general state space Markov chains with stationary transition probabilities. Replacing the assumption of reversibility with a weaker assumption, we still obtain a simple necessary and sufficient condition for the spectral gap property of the associated Markov operator in terms of the isoperimetric constant. We show that this result can be applied to a large class of Markov chains, including those that are related to positive recurrent finite-range random walks on Z.

Archimedean copulas, exchangeability, and max-stability

1994 ◽  
Vol 31 (2) ◽  
pp. 383-390 ◽  
Author(s):  
Rocco Ballerini
Keyword(s):  
Random Variables ◽  
Distribution Functions ◽  
Archimedean Copula ◽  
Archimedean Copulas ◽  
Monotone Transformation ◽  
Stable Property ◽  
Finite Dimensional ◽  
Finite Dimensional Distribution ◽  
Exchangeable Sequence ◽  
Dimensional Distribution

An exchangeable sequence of random variables is constructed with all finite-dimensional distribution functions having an Archimedean copula (as defined by Schweizer and Sklar (1983)). Through a monotone transformation of this exchangeable sequence, we obtain and characterize the class of exchangeable sequences possessing the max-stable property as defined by De Haan and Rachev (1989). Several parametric examples are given.

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