Asymptotic behaviour of sample weighted circuits representing recurrent Markov chains

1990 ◽  
Vol 27 (03) ◽  
pp. 545-556 ◽  
Author(s):  
S. Kalpazidou
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Asymptotic Behaviour ◽  
Probabilistic Interpretation ◽  
Stationary Markov Chain

The asymptotic behaviour of the sequence (𝒞 n (ω), wc,n (ω)/n), is studied where 𝒞 n (ω) is the class of all cycles c occurring along the trajectory ωof a recurrent strictly stationary Markov chain (ξ n ) until time n and wc,n (ω) is the number of occurrences of the cycle c until time n. The previous sequence of sample weighted classes converges almost surely to a class of directed weighted cycles (𝒞∞, ω c ) which represents uniquely the chain (ξ n ) as a circuit chain, and ω c is given a probabilistic interpretation.

Asymptotic behaviour of sample weighted circuits representing recurrent Markov chains

1990 ◽  
Vol 27 (3) ◽  
pp. 545-556 ◽  
Author(s):  
S. Kalpazidou
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Asymptotic Behaviour ◽  
Probabilistic Interpretation ◽  
Stationary Markov Chain

The asymptotic behaviour of the sequence (𝒞n(ω), wc,n(ω)/n), is studied where 𝒞n(ω) is the class of all cycles c occurring along the trajectory ωof a recurrent strictly stationary Markov chain (ξ n) until time n and wc,n(ω) is the number of occurrences of the cycle c until time n. The previous sequence of sample weighted classes converges almost surely to a class of directed weighted cycles (𝒞∞, ωc) which represents uniquely the chain (ξ n) as a circuit chain, and ω c is given a probabilistic interpretation.

Exponential ergodicity in derived Markov chains

1968 ◽  
Vol 5 (03) ◽  
pp. 669-678 ◽  
Author(s):  
Jozef L. Teugels
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Continuous Time ◽  
Exponential Ergodicity ◽  
Stationary Markov Chain ◽  
Continuous Time Markov Chains ◽  
General Proposition

A general proposition is proved stating that the exponential ergodicity of a stationary Markov chain is preserved for derived Markov chains as defined by Cohen [2], [3]. An application to a certain type of continuous time Markov chains is included.

Markov Chains in Many Dimensions

1994 ◽  
Vol 26 (3) ◽  
pp. 756-774 ◽  
Author(s):  
Dimitris N. Politis
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Maximum Entropy ◽  
Stationary Distribution ◽  
Random Fields ◽  
Special Class ◽  
Markov Random Fields ◽  
One Dimension ◽  
Stationary Markov Chain ◽  
Markov Random

A generalization of the notion of a stationary Markov chain in more than one dimension is proposed, and is found to be a special class of homogeneous Markov random fields. Stationary Markov chains in many dimensions are shown to possess a maximum entropy property, analogous to the corresponding property for Markov chains in one dimension. In addition, a representation of Markov chains in many dimensions is provided, together with a method for their generation that converges to their stationary distribution.

Extreme values, range and weak convergence of integrals of Markov chains

1982 ◽  
Vol 19 (02) ◽  
pp. 272-288 ◽  
Author(s):  
P. J. Brockwell ◽  
S. I. Resnick ◽  
N. Pacheco-Santiago
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
Markov Chains ◽  
Asymptotic Behaviour ◽  
State Space ◽  
Extreme Values ◽  
Finite State ◽  
Integral Process ◽  
Maximum Minimum ◽  
Finite State Space

A study is made of the maximum, minimum and range on [0,t] of the integral processwhereSis a finite state-space Markov chain. Approximate results are derived by establishing weak convergence of a sequence of such processes to a Wiener process. For a particular family of two-state stationary Markov chains we show that the corresponding centered integral processes exhibit the Hurst phenomenon to a remarkable degree in their pre-asymptotic behaviour.

Poisson limit law for Markov chains

1991 ◽  
Vol 11 (3) ◽  
pp. 501-513 ◽  
Author(s):  
B. Pitskel
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Stationary Markov Chain ◽  
Poisson Limit

AbstractFor a mixing stationary Markov chain we prove a Poisson limit law for the recurrence to small cylindrical sets. Since hyperbolic torus automorphisms are Markov chains, the result carries over to these transformations.

Exponential ergodicity in derived Markov chains

1968 ◽  
Vol 5 (3) ◽  
pp. 669-678 ◽  
Author(s):  
Jozef L. Teugels
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Continuous Time ◽  
Exponential Ergodicity ◽  
Stationary Markov Chain ◽  
Continuous Time Markov Chains ◽  
General Proposition

A general proposition is proved stating that the exponential ergodicity of a stationary Markov chain is preserved for derived Markov chains as defined by Cohen [2], [3]. An application to a certain type of continuous time Markov chains is included.

Representation of denumerable Markov chains with multiple states by weighted circuits

1989 ◽  
Vol 26 (1) ◽  
pp. 23-35 ◽  
Author(s):  
S. Kalpazidou
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Electrical Current ◽  
Constructive Solution ◽  
Stationary Markov Chain ◽  
Planar Network ◽  
Countable Infinity ◽  
Physical Laws

The constructive solution to the problem of representing a strictly stationary Markov chainζwith a countable infinity ofr-sequences (i1,i2, · ··,ir),r> 1, as states by a class of directed weighted circuits is given. Associating the chainζwith its dual chainηhaving reversed states and the same transition law, a connection with physical laws that govern diffusion of electrical current through a directed planar network withr-series-connected nodes is shown.

Markov Chains in Many Dimensions

1994 ◽  
Vol 26 (03) ◽  
pp. 756-774 ◽  
Author(s):  
Dimitris N. Politis
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Maximum Entropy ◽  
Stationary Distribution ◽  
Random Fields ◽  
Special Class ◽  
Markov Random Fields ◽  
One Dimension ◽  
Stationary Markov Chain ◽  
Markov Random

A generalization of the notion of a stationary Markov chain in more than one dimension is proposed, and is found to be a special class of homogeneous Markov random fields. Stationary Markov chains in many dimensions are shown to possess a maximum entropy property, analogous to the corresponding property for Markov chains in one dimension. In addition, a representation of Markov chains in many dimensions is provided, together with a method for their generation that converges to their stationary distribution.

Extreme values, range and weak convergence of integrals of Markov chains

1982 ◽  
Vol 19 (2) ◽  
pp. 272-288 ◽  
Author(s):  
P. J. Brockwell ◽  
S. I. Resnick ◽  
N. Pacheco-Santiago
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
Markov Chains ◽  
Asymptotic Behaviour ◽  
State Space ◽  
Extreme Values ◽  
Finite State ◽  
Integral Process ◽  
Maximum Minimum ◽  
Finite State Space

A study is made of the maximum, minimum and range on [0, t] of the integral process where S is a finite state-space Markov chain. Approximate results are derived by establishing weak convergence of a sequence of such processes to a Wiener process. For a particular family of two-state stationary Markov chains we show that the corresponding centered integral processes exhibit the Hurst phenomenon to a remarkable degree in their pre-asymptotic behaviour.

XXIV.—Conditions for the Ergodicity of Non-homogeneous Finite Markov Chains

1957 ◽  
Vol 64 (4) ◽  
pp. 369-380 ◽  
Author(s):  
J. L. Mott
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Asymptotic Behaviour ◽  
Transition Probabilities ◽  
Finite Markov Chain ◽  
Finite Markov Chains ◽  
Product Of Matrices

SynopsisIn this note we study the asymptotic behaviour of a product of matrices where Pj is a matrix of transition probabilities in a non-homogeneous finite Markov chain. We give conditions that (i) the rows of P(n) tend to identity and that (ii) P(n) tends to a limit matrix with identical rows.

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