Markov Chains in Many Dimensions

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.

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Mathematics and Innovation in Engineering

Key Engineering Materials ◽

10.4028/www.scientific.net/kem.380.3 ◽

2008 ◽

Vol 380 ◽

pp. 3-14

Author(s):

Elena Beretta ◽

Alberto Gandolfi ◽

C.C.A. Sastri

Keyword(s):

Markov Chains ◽

Large Deviations ◽

Random Fields ◽

Spin Glasses ◽

Markov Random Fields ◽

Imaging Technologies ◽

Inverse Conductivity Problem ◽

Markov Random ◽

The Way ◽

Conductivity Problem

We present some examples of mathematical discoveries whose original import was mainly theoretical but which later ended up triggering extraordinary ad- vances in engineering, sometimes all the way down to technological realizations and market products. The examples we cite include Markov chains and Markov random fields, spin glasses, large deviations and the inverse conductivity problem, and their effects in various areas such as communication and imaging technologies.

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Periodic Ornstein-Uhlenbeck processes driven by Lévy processes

Journal of Applied Probability ◽

10.1017/s0021900200022014 ◽

2002 ◽

Vol 39 (04) ◽

pp. 748-763 ◽

Cited By ~ 1

Author(s):

Jan Pedersen

Keyword(s):

Stationary Distribution ◽

Random Fields ◽

Markov Random Fields ◽

Lévy Processes ◽

Levy Processes ◽

Uhlenbeck Process ◽

Stationary Distributions ◽

Markov Random ◽

Ornstein Uhlenbeck Process

In this paper, the class of periodic Ornstein-Uhlenbeck processes is defined. It is shown that periodic Ornstein-Uhlenbeck processes are stationary Markov random fields and the class of stationary distributions is characterized. In particular, any self-decomposable distribution is the stationary distribution of some periodic Ornstein-Uhlenbeck process. As examples, gamma periodic Ornstein-Uhlenbeck processes and Gaussian periodic Ornstein-Uhlenbeck processes are considered.

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Periodic Ornstein-Uhlenbeck processes driven by Lévy processes

Journal of Applied Probability ◽

10.1239/jap/1037816016 ◽

2002 ◽

Vol 39 (4) ◽

pp. 748-763 ◽

Cited By ~ 6

Author(s):

Jan Pedersen

Keyword(s):

Stationary Distribution ◽

Random Fields ◽

Markov Random Fields ◽

Lévy Processes ◽

Levy Processes ◽

Uhlenbeck Process ◽

Stationary Distributions ◽

Markov Random ◽

Ornstein Uhlenbeck Process

In this paper, the class of periodic Ornstein-Uhlenbeck processes is defined. It is shown that periodic Ornstein-Uhlenbeck processes are stationary Markov random fields and the class of stationary distributions is characterized. In particular, any self-decomposable distribution is the stationary distribution of some periodic Ornstein-Uhlenbeck process. As examples, gamma periodic Ornstein-Uhlenbeck processes and Gaussian periodic Ornstein-Uhlenbeck processes are considered.

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