scholarly journals Large deviations for the skew-detailed-balance lifted-Markov processes to sample the equilibrium distribution of the Curie–Weiss model

2021 ◽  
Vol 2021 (10) ◽  
pp. 103202
Author(s):  
Cécile Monthus
Keyword(s):  
Large Deviations ◽  
Markov Processes ◽  
Equilibrium Distribution ◽  
Detailed Balance

scholarly journals A Second Law for Open Markov Processes

2016 ◽  
Vol 23 (01) ◽  
pp. 1650006 ◽  
Author(s):  
Blake S. Pollard
Keyword(s):  
Free Energy ◽  
Markov Process ◽  
Markov Processes ◽  
Relative Entropy ◽  
Upper Bound ◽  
Equilibrium Distribution ◽  
Detailed Balance ◽  
Rate Of Change ◽  
Second Law ◽  
Boundary States

In this paper we define the notion of an open Markov process. An open Markov process is a generalization of an ordinary Markov process in which populations are allowed to flow in and out of the system at certain boundary states. We show that the rate of change of relative entropy in an open Markov process is less than or equal to the flow of relative entropy through its boundary states. This can be viewed as a generalization of the Second Law for open Markov processes. In the case of a Markov process whose equilibrium obeys detailed balance, this inequality puts an upper bound on the rate of change of the free energy for any non-equilibrium distribution.

Coherent random walks arising in some genetical models

1976 ◽  
Vol 351 (1664) ◽  
pp. 19-31 ◽  
Keyword(s):  
Population Genetics ◽  
Random Walks ◽  
Markov Processes ◽  
Stochastic Models ◽  
Equilibrium Distribution ◽  
Large Sample ◽  
Random Variation

Certain stochastic models used in population genetics have the form of Markov processes in which a group of N points moves randomly on a line, and in which an equilibrium distribution exists for the relative configura­tion of the group. The properties of this equilibrium are studied, with particular reference to a certain limiting situation as N becomes large. In this limit the group of points is distributed like a large sample from a distribution which is itself subject to random variation.

Relaxed Markov processes

1983 ◽  
Vol 15 (04) ◽  
pp. 769-782 ◽  
Author(s):  
P. Whittle
Keyword(s):  
Markov Process ◽  
Markov Processes ◽  
Equilibrium Distribution ◽  
Closed Model ◽  
Jackson Networks ◽  
The Creation

The concept of relaxing a Markov process is introduced; this is the creation of additional transitions between ergodic classes of the process in such a way as to conserve the existing equilibrium distribution within ergodic classes. The ‘open' version of a ‘closed' model of migration, polymerisation etc. often has this character. As further examples, generalized versions of Jackson networks and networks with clustering nodes are given.

scholarly journals A Note on Rényi's ‘Record’ Problem and Engel's Series

2018 ◽  
Vol 61 (2) ◽  
pp. 363-369 ◽  
Author(s):  
Lulu Fang ◽  
Min Wu
Keyword(s):  
Number Theory ◽  
Large Deviations ◽  
Markov Processes ◽  
Classical Limit ◽  
Limit Theorems ◽  
London Math ◽  
Law Of Large Numbers ◽  
Record Times ◽  
Iterated Logarithm ◽  
Large Numbers

AbstractIn 1973, Williams [D. Williams, On Rényi's ‘record’ problem and Engel's series, Bull. London Math. Soc.5 (1973), 235–237] introduced two interesting discrete Markov processes, namely C-processes and A-processes, which are related to record times in statistics and Engel's series in number theory respectively. Moreover, he showed that these two processes share the same classical limit theorems, such as the law of large numbers, central limit theorem and law of the iterated logarithm. In this paper, we consider the large deviations for these two Markov processes, which indicate that there is a difference between C-processes and A-processes in the context of large deviations.

scholarly journals A compensation approach for two-dimensional Markov processes

1993 ◽  
Vol 25 (4) ◽  
pp. 783-817 ◽  
Author(s):  
I. J. B. F. Adan ◽  
J. Wessels ◽  
W. H. M. Zijm
Keyword(s):  
Random Walks ◽  
Markov Processes ◽  
Infinite Series ◽  
Equilibrium Distribution ◽  
Two Dimensional ◽  
Compensation Approach ◽  
Queueing Processes ◽  
Random Behaviour ◽  
Elegant Solution

Several queueing processes may be modeled as random walks on a multidimensional grid. In this paper the equilibrium distribution for the case of a two-dimensional grid is considered. In previous research it has been shown that for some two-dimensional random walks the equilibrium distribution has the form of an infinite series of products of powers which can be constructed with a compensation procedure. The object of the present paper is to investigate under which conditions such an elegant solution exists and may be found with a compensation approach. The conditions can be easily formulated in terms of the random behaviour in the inner area and the drift on the boundaries.

Extended continued fractions, recurrence relations and two-dimensional Markov processes

1989 ◽  
Vol 21 (2) ◽  
pp. 357-375 ◽  
Author(s):  
C. E. M. Pearce
Keyword(s):  
Markov Processes ◽  
Continued Fractions ◽  
Recurrence Relations ◽  
Equilibrium Distribution ◽  
Linear Recurrence ◽  
Natural State ◽  
Two Dimensional ◽  
State Spaces ◽  
General Order ◽  
Dimension 2

Connections between Markov processes and continued fractions have long been known (see, for example, Good [8]). However the usefulness of extended continued fractions in such a context appears not to have been explored. In this paper a convergence theorem is established for a class of extended continued fractions and used to provide well-behaved solutions for some general order linear recurrence relations such as arise in connection with the equilibrium distribution of state for some Markov processes whose natural state spaces are of dimension 2. Specific application is made to a multiserver version of a queueing problem studied by Neuts and Ramalhoto [13] and to a model proposed by Cohen [5] for repeated call attempts in teletraffic.

Sign in / Sign up

Export Citation Format

Share Document