Stationary distribution of finite-size systems with absorbing states

2005 ◽  
Vol 72 (2) ◽  
Author(s):  
Tânia Tomé ◽  
Mário J. de Oliveira
Keyword(s):  
Stationary Distribution ◽  
Finite Size ◽  
Absorbing States

AN INVESTIGATION OF THE PHASE TRANSITIONS OF A FAMILY OF PROBABILISTIC AUTOMATA

2004 ◽  
Vol 07 (01) ◽  
pp. 93-123
Author(s):  
HEINZ MÜHLENBEIN ◽  
THOMAS AUS DER FÜNTEN
Keyword(s):  
Phase Transitions ◽  
Stationary Distribution ◽  
Classification Scheme ◽  
Mean Field ◽  
Finite Size ◽  
Order Parameters ◽  
Finite Size Scaling ◽  
Probabilistic Cellular Automata ◽  
Probabilistic Automata ◽  
Major Transitions

We investigate a family of totalistic probabilistic cellular automata (PCA) which depend on three parameters. For the uniform random neighborhood and for the symmetric 1D PCA the exact stationary distribution is computed for all finite n. This result is used to evaluate approximations (uni-variate and bi-variate marginals). It is proven that the uni-variate approximation (also called mean-field) is exact for the uniform random neighborhood PCA. The exact results and the approximations are used to investigate phase transitions. We compare the results of two order parameters, the uni-variate marginal and the normalized entropy. Sometimes different transitions are indicated by the Ehrenfest classification scheme. This result shows the limitations of using just one or two order parameters for detecting and classifying major transitions of the stationary distribution. Furthermore, finite size scaling is investigated. We show that extrapolations to n=∞ from numerical calculations of finite n can be misleading in difficult parameter regions. Here, exact analytical estimates are necessary.

Random Walks in the Presence of Absorbing Barriers in Parrondo's Games

2014 ◽  
Vol 14 (01) ◽  
pp. 1550003 ◽  
Author(s):  
Liu Yang ◽  
Kai-Xuan Zheng ◽  
Neng-Gang Xie ◽  
Ye Ye ◽  
Lu Wang
Keyword(s):  
Markov Chains ◽  
Random Walks ◽  
Stationary Distribution ◽  
Discrete Time ◽  
Transition Matrix ◽  
Multi Agent ◽  
Probability Transition Matrix ◽  
Initial States ◽  
Absorbing States

For a multi-agent spatial Parrondo's model that it is composed of games A and B, we use the discrete time Markov chains to derive the probability transition matrix. Then, we respectively deduce the stationary distribution for games A and B played individually and the randomized combination of game A + B. We notice that under a specific set of parameters, two absorbing states instead of a fixed stationary distribution exist in game B. However, the randomized game A + B can jump out of the absorbing states of game B and has a fixed stationary distribution because of the "agitating" role of game A. Moreover, starting at different initial states, we deduce the probabilities of absorption at the two absorbing barriers.

scholarly journals On the quasi-stationary distribution for queueing networks with defective routing

1997 ◽  
Vol 38 (4) ◽  
pp. 454-463 ◽  
Author(s):  
Richard J. Boucherie
Keyword(s):  
Markov Chains ◽  
Stationary Distribution ◽  
Discrete Time ◽  
Queueing Networks ◽  
Product Form ◽  
Stationary Distributions ◽  
Absorbing States ◽  
Quasi Stationary Distribution

AbstractThis note introduces quasi-local-balance for discrete-time Markov chains with absorbing states. From quasi-local-balance product-form quasi-stationary distributions are derived by analogy with product-form stationary distributions for Markov chains that satisfy local balance.

SIEGMUND DUALITY FOR MARKOV CHAINS ON PARTIALLY ORDERED STATE SPACES

2017 ◽  
Vol 32 (4) ◽  
pp. 495-521 ◽  
Author(s):  
Paweł Lorek
Keyword(s):  
Stationary Distribution ◽  
The Other ◽  
Stochastic Monotonicity ◽  
State Spaces ◽  
Partially Ordered ◽  
A Chain ◽  
Gambler's Ruin Problem ◽  
Absorbing States ◽  
Ordered State ◽  
Absorption Probabilities

For a Markov chain on a finite partially ordered state space, we show that its Siegmund dual exists if and only if the chain is Möbius monotone. This is an extension of Siegmund's result for totally ordered state spaces, in which case the existence of the dual is equivalent to the usual stochastic monotonicity. Exploiting the relation between the stationary distribution of an ergodic chain and the absorption probabilities of its Siegmund dual, we present three applications: calculating the absorption probabilities of a chain with two absorbing states knowing the stationary distribution of the other chain; calculating the stationary distribution of an ergodic chain knowing the absorption probabilities of the other chain; and providing a stable simulation scheme for the stationary distribution of a chain provided we can simulate its Siegmund dual. These are accompanied by concrete examples: the gambler's ruin problem with arbitrary winning/losing probabilities; a non-symmetric game; an extension of a birth and death chain; a chain corresponding to the Fisher–Wright model; a non-standard tandem network of two servers, and the Ising model on a circle. We also show that one can construct a strong stationary dual chain by taking the appropriate Doob transform of the Siegmund dual of the time-reversed chain.

Classification and lumpability in the stochastic Hopfield model

2001 ◽  
Vol 33 (4) ◽  
pp. 930-943 ◽  
Author(s):  
R. L. Paige
Keyword(s):  
Markov Chain ◽  
Stationary Distribution ◽  
Energy Function ◽  
Hopfield Model ◽  
Zero Temperature ◽  
Absorbing States ◽  
Classification Time

Connections between classification and lumpability in the stochastic Hopfield model (SHM) are explored and developed. A simplification of the SHM's complexity based upon its inherent lumpability is derived. Contributions resulting from this reduction in complexity include: (i) computationally feasible classification time computations; (ii) a development of techniques for enumerating the stationary distribution of the SHM's energy function; and (iii) a characterization of the set of possible absorbing states of the Markov chain associated with the zero temperature SHM.

Classification and lumpability in the stochastic Hopfield model

2001 ◽  
Vol 33 (04) ◽  
pp. 930-943 ◽  
Author(s):  
R. L. Paige
Keyword(s):  
Markov Chain ◽  
Stationary Distribution ◽  
Energy Function ◽  
Hopfield Model ◽  
Zero Temperature ◽  
Absorbing States ◽  
Classification Time

Connections between classification and lumpability in the stochastic Hopfield model (SHM) are explored and developed. A simplification of the SHM's complexity based upon its inherent lumpability is derived. Contributions resulting from this reduction in complexity include: (i) computationally feasible classification time computations; (ii) a development of techniques for enumerating the stationary distribution of the SHM's energy function; and (iii) a characterization of the set of possible absorbing states of the Markov chain associated with the zero temperature SHM.

scholarly journals Dirichlet approximation of equilibrium distributions in Cannings models with mutation

2017 ◽  
Vol 49 (3) ◽  
pp. 927-959
Author(s):  
Han L. Gan ◽  
Adrian Röllin ◽  
Nathan Ross
Keyword(s):  
Stationary Distribution ◽  
Finite Population ◽  
Dirichlet Distribution ◽  
Finite Size ◽  
First Case ◽  
Probabilistic Description ◽  
New Development ◽  
Equilibrium Distributions ◽  
Haploid Population ◽  
Independent Mutation

AbstractConsider a haploid population of fixed finite size with a finite number of allele types and having Cannings exchangeable genealogy with neutral mutation. The stationary distribution of the Markov chain of allele counts in each generation is an important quantity in population genetics but has no tractable description in general. We provide upper bounds on the distributional distance between the Dirichlet distribution and this finite-population stationary distribution for the Wright–Fisher genealogy with general mutation structure and the Cannings exchangeable genealogy with parent independent mutation structure. In the first case, the bound is small if the population is large and the mutations do not depend too much on parent type; 'too much' is naturally quantified by our bound. In the second case, the bound is small if the population is large and the chance of three-mergers in the Cannings genealogy is small relative to the chance of two-mergers; this is the same condition to ensure convergence of the genealogy to Kingman's coalescent. These results follow from a new development of Stein's method for the Dirichlet distribution based on Barbour's generator approach and a probabilistic description of the semigroup of the Wright–Fisher diffusion due to Griffiths and Li (1983) and Tavaré (1984).

Normal approximations to discrete unimodal distributions

1986 ◽  
Vol 23 (04) ◽  
pp. 1013-1018
Author(s):  
B. G. Quinn ◽  
H. L. MacGillivray
Keyword(s):  
Stationary Distribution ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Hypergeometric Distribution ◽  
Death Process ◽  
Normal Approximations ◽  
Discrete Random Variables ◽  
Birth Death

Sufficient conditions are presented for the limiting normality of sequences of discrete random variables possessing unimodal distributions. The conditions are applied to obtain normal approximations directly for the hypergeometric distribution and the stationary distribution of a special birth-death process.

The partial realization theory of finite size two-dimensional arrays

1981 ◽  
Vol 64 (10) ◽  
pp. 1-8
Author(s):  
Tsuyoshi Matsuo ◽  
Yasumichi Hasegawa ◽  
Yoshikuni Okada
Keyword(s):  
Finite Size ◽  
Two Dimensional ◽  
Realization Theory
Sign in / Sign up

Export Citation Format

Share Document