On Martin Boundaries for the Direct Product of Markov Chains

1967 ◽  
Vol 12 (2) ◽  
pp. 307-310 ◽  
Author(s):  
S. A. Molchanov
Keyword(s):  
Markov Chains ◽  
Direct Product ◽  
Martin Boundaries

On Finite Coding Factors of a Class of Random Markov Chains

1994 ◽  
Vol 37 (3) ◽  
pp. 399-407 ◽  
Author(s):  
M. Rahe
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Direct Product ◽  
Finite Length ◽  
Bernoulli Shift ◽  
Finite Rotation ◽  
Step Size ◽  
Zero Entropy

AbstractFor k-step Markov chains, factors generated by finite length codes split off with Bernoulli complement when maximal in entropy. Those not maximal are relatively finite in another factor which generates or splits off.These results extend to random Markov chains with finite expected step size, implying that random Markov chains with finite expected step size can have only finitely many ergodic components, each of which is isomorphic to a finite rotation, a Bernoulli shift, or a direct product of a Bernoulli shift with a finite rotation. This result limits the type of zero entropy factors which occur in random Markov chains with finite expected step size, providing a counterpoint to the work of Kalikow, Katznelson, and Weiss, who have shown that each zero entropy process can be embedded in some random Markov chain.Extending Rudolph and Schwarz, random Markov chains with finite expected step size are limits in of their canonical Markov approximants. The -closure of the class is the Bernoulli cross Generalized Von Neuman processes.Finitary isomorphism of aperiodic ergodic random Markov chains with finite expected step size is considered.Applications are made to a class of generalized baker's transformations.

scholarly journals Passage-Time Moments for Positively Recurrent Markov Chains

2001 ◽  
Vol 162 ◽  
pp. 169-185
Author(s):  
Tokuzo Shiga ◽  
Akinobu Shimizu ◽  
Takahiro Soshi
Keyword(s):  
Markov Chain ◽  
Convergence Rate ◽  
Markov Chains ◽  
Direct Product ◽  
Transition Probability ◽  
Passage Time ◽  
State Spaces ◽  
Countable State ◽  
Fractional Moments ◽  
Passage Times

Fractional moments of the passage-times are considered for positively recurrent Markov chains with countable state spaces. A criterion of the finiteness of the fractional moments is obtained in terms of the convergence rate of the transition probability to the stationary distribution. As an application it is proved that the passage time of a direct product process of Markov chains has the same order of the fractional moments as that of the single Markov chain.

scholarly journals Substitution Markov chains and Martin boundaries

2016 ◽  
Vol 46 (6) ◽  
pp. 1963-1985
Author(s):  
David Koslicki ◽  
Manfred Denker
Keyword(s):  
Markov Chains ◽  
Martin Boundaries

scholarly journals Martin boundaries for certain Markov chains

1963 ◽  
Vol 15 (2) ◽  
pp. 113-128 ◽  
Author(s):  
J. LAMPERTI ◽  
J. L. SNELL
Keyword(s):  
Markov Chains ◽  
Martin Boundaries

scholarly journals Martin boundaries of cartesian products of markov chains

1992 ◽  
Vol 128 ◽  
pp. 153-169 ◽  
Author(s):  
Massimo A. Picardello ◽  
Wolfgang Woess
Keyword(s):  
Markov Chains ◽  
Harmonic Functions ◽  
Cartesian Product ◽  
Martin Boundary ◽  
State Spaces ◽  
Cartesian Products ◽  
Discrete State ◽  
Stochastic Transition ◽  
Transition Operators ◽  
Martin Boundaries

Let P and Q be the stochastic transition operators of two time-homogeneous, irreducible Markov chains with countable, discrete state spaces X and Y, respectively. On the Cartesian product Z = X x Y, define a transition operator of the form Ra = a·P + (1 — a) · Q, 0 < a < 1, where P is considered to act on the first variable and Q on the second. The principal purpose of this paper is to describe the minimal Martin boundary of Ra (consisting of the minimal positive eigenfunctions of Ra with respect to some eigenvalue t, also called t-harmonic functions) in terms of the minimal Martin boundaries of P and Q.

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