Towards consensus: some convergence theorems on repeated averaging

1977 ◽  
Vol 14 (1) ◽  
pp. 89-97 ◽  
Author(s):  
S. Chatterjee ◽  
E. Seneta
Keyword(s):  
Markov Chain ◽  
Consensus Problem ◽  
Stochastic Matrices ◽  
Convergence Theorems ◽  
Strong Ergodicity ◽  
Markov Chain Theory ◽  
Inhomogeneous Markov Chain ◽  
Chain Theory ◽  
Equivalent Conditions

The problem of tendency to consensus in an information-exchanging operation is connected with the ergodicity problem for backwards products of stochastic matrices. For such products, weak and strong ergodicity, defined analogously to these concepts for forward products of inhomogeneous Markov chain theory, are shown (in contrast to that theory) to be equivalent. Conditions for ergodicity are derived and their relation to the consensus problem is considered.

Towards consensus: some convergence theorems on repeated averaging

1977 ◽  
Vol 14 (01) ◽  
pp. 89-97 ◽  
Author(s):  
S. Chatterjee ◽  
E. Seneta
Keyword(s):  
Markov Chain ◽  
Consensus Problem ◽  
Stochastic Matrices ◽  
Convergence Theorems ◽  
Strong Ergodicity ◽  
Markov Chain Theory ◽  
Inhomogeneous Markov Chain ◽  
Chain Theory ◽  
Equivalent Conditions

The problem of tendency to consensus in an information-exchanging operation is connected with the ergodicity problem for backwards products of stochastic matrices. For such products, weak and strong ergodicity, defined analogously to these concepts for forward products of inhomogeneous Markov chain theory, are shown (in contrast to that theory) to be equivalent. Conditions for ergodicity are derived and their relation to the consensus problem is considered.

scholarly journals NONSTATIONARY MIXING AND THE UNIQUE ERGODICITY OF ADIC TRANSFORMATIONS

2009 ◽  
Vol 09 (03) ◽  
pp. 335-391 ◽  
Author(s):  
ALBERT M. FISHER
Keyword(s):  
Markov Chain ◽  
Dynamical Systems ◽  
Finite Type ◽  
Weak Ergodicity ◽  
Subshift Of Finite Type ◽  
Topological Mixing ◽  
Markov Chain Theory ◽  
Inhomogeneous Markov Chain ◽  
Chain Theory ◽  
Uniquely Ergodic

We define topological and measure-theoretic mixing for nonstationary dynamical systems and prove that for a nonstationary subshift of finite type, topological mixing implies the minimality of any adic transformation defined on the edge space, while if the Parry measure sequence is mixing, the adic transformation is uniquely ergodic. We also show this measure theoretic mixing is equivalent to weak ergodicity of the edge matrices in the sense of inhomogeneous Markov chain theory.

Finite non-homogeneous Markov chains: Asymptotic behaviour

1976 ◽  
Vol 8 (3) ◽  
pp. 502-516 ◽  
Author(s):  
Harry Cohn
Keyword(s):  
Markov Chain ◽  
Asymptotic Behaviour ◽  
Transition Probabilities ◽  
Homogeneous Markov Chain ◽  
Stochastic Matrices ◽  
Weak Ergodicity ◽  
General Picture ◽  
Markov Chain Theory ◽  
Chain Theory ◽  
Probabilistic Nature

The paper is concerned with aspects of the behaviour of the products of finite stochastic matrices, the methods used in the proofs being of a probabilistic nature. The main result of the paper (Theorem 1) presents a general picture of the asymptotic behaviour of the transition probabilities between various groups of states. A unified treatment of some results of non-homogeneous Markov chain theory pertaining to weak ergodicity is then given.

Finite non-homogeneous Markov chains: Asymptotic behaviour

1976 ◽  
Vol 8 (03) ◽  
pp. 502-516 ◽  
Author(s):  
Harry Cohn
Keyword(s):  
Markov Chain ◽  
Asymptotic Behaviour ◽  
Transition Probabilities ◽  
Homogeneous Markov Chain ◽  
Stochastic Matrices ◽  
Weak Ergodicity ◽  
General Picture ◽  
Markov Chain Theory ◽  
Chain Theory ◽  
Probabilistic Nature

The paper is concerned with aspects of the behaviour of the products of finite stochastic matrices, the methods used in the proofs being of a probabilistic nature. The main result of the paper (Theorem 1) presents a general picture of the asymptotic behaviour of the transition probabilities between various groups of states. A unified treatment of some results of non-homogeneous Markov chain theory pertaining to weak ergodicity is then given.

A note on simultaneous recurrence conditions on a set of denumerable stochastic matrices

1978 ◽  
Vol 15 (4) ◽  
pp. 842-847 ◽  
Author(s):  
A. Federgruen ◽  
A. Hordijk ◽  
H. C. Tijms
Keyword(s):  
Markov Chain ◽  
Stochastic Control ◽  
Metric Space ◽  
Considerable Extent ◽  
Control Problems ◽  
Stochastic Matrices ◽  
Compact Metric Space ◽  
Stochastic Control Problems ◽  
Markov Chain Theory ◽  
Chain Theory

In this paper we consider a set of denumerable stochastic matrices where the parameter set is a compact metric space. We give a number of simultaneous recurrence conditions on the stochastic matrices and establish equivalences between these conditions. The results obtained generalize corresponding results in Markov chain theory to a considerable extent and have applications in stochastic control problems.

A note on simultaneous recurrence conditions on a set of denumerable stochastic matrices

1978 ◽  
Vol 15 (04) ◽  
pp. 842-847 ◽  
Author(s):  
A. Federgruen ◽  
A. Hordijk ◽  
H. C. Tijms
Keyword(s):  
Markov Chain ◽  
Stochastic Control ◽  
Metric Space ◽  
Considerable Extent ◽  
Control Problems ◽  
Stochastic Matrices ◽  
Compact Metric Space ◽  
Stochastic Control Problems ◽  
Markov Chain Theory ◽  
Chain Theory

In this paper we consider a set of denumerable stochastic matrices where the parameter set is a compact metric space. We give a number of simultaneous recurrence conditions on the stochastic matrices and establish equivalences between these conditions. The results obtained generalize corresponding results in Markov chain theory to a considerable extent and have applications in stochastic control problems.

scholarly journals Generation method for the PV power time series combining the decomposition technique and Markov chain theory

2017 ◽  
Vol 2017 (13) ◽  
pp. 2026-2031
Author(s):  
Shenzhi Xu ◽  
Xiaomeng Ai ◽  
Jiakun Fang ◽  
Jinyu Wen ◽  
Pai Li ◽  
...  
Keyword(s):  
Time Series ◽  
Markov Chain ◽  
Decomposition Technique ◽  
Markov Chain Theory ◽  
Chain Theory

scholarly journals An ergodic algorithm for generating knots with a prescribed injectivity radius

2019 ◽  
Vol 28 (06) ◽  
pp. 1950045
Author(s):  
Kyle Leland Chapman
Keyword(s):  
Markov Chain ◽  
Lattice Model ◽  
Probability Distributions ◽  
Positive Probability ◽  
Bending Angle ◽  
Markov Chain Theory ◽  
Countable Space ◽  
Chain Theory ◽  
Random Reflections ◽  
Planar Polygon

The first provably ergodic algorithm for sampling the space of thick equilateral knots off-lattice, as a function of thickness, will be described. This algorithm is based on previous algorithms of applying random reflections. It is an off-lattice generalization of the pivot algorithm. This move to an off-lattice model provides a huge improvement in power and efficacy in that samples can have arbitrary values for parameters such as the thickness constraint, bending angle, and torsion, while the lattice forces these parameters into a small number of specific values. This benefit requires working in a manifold rather than a finite or countable space, which forces the use of more novel methods in Markov–Chain theory. To prove the validity of the algorithm, we describe a method for turning any knot into the regular planar polygon using only thickness non-decreasing moves. This approach ensures that the algorithm has a positive probability of connecting any two knots with the required thickness constraint which is used to show that the algorithm is ergodic. This ergodic sampling allows for a statistically valid method for estimating probability distributions of arbitrary functions on the space of thick knots.

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