The Existence of Stationary Measures for Certain Markov Processes
01 Jan 1956-pp 113-124
TL;DR: In this article, it is shown that a certain type of recurrence condition implies the existence of a possible infinite stationary measure, which is a special case of a stationary distribution for Markov process.
Abstract: : The idea of a stationary distribution for a Markov process can be extended to include 'distribution' or measures which are infinite. It is shown that a certain type of recurrence condition implies the existence of a possible infinite stationary measure. Applications are discussed.
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01 Jan 1990TL;DR: In this paper, a comprehensive introduction to probability theory covering laws of large numbers, central limit theorem, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion is presented.
Abstract: This book is an introduction to probability theory covering laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a comprehensive treatment concentrating on the results that are the most useful for applications. Its philosophy is that the best way to learn probability is to see it in action, so there are 200 examples and 450 problems.
5,168 citations
TL;DR: In this paper, an update of, and a supplement to, a 1986 survey paper by the author on basic properties of strong mixing conditions is presented, which is an extension to the survey paper.
Abstract: This is an update of, and a supplement to, a 1986 survey paper by the author on basic
properties of strong mixing conditions.
908 citations
Cites background from "The Existence of Stationary Measure..."
...The (strictly stationary) Markov chain X is said to be “Harris recurrent” [81] if the following holds for μ–a....
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01 Jun 1985
TL;DR: A survey of the properties of strong mixing conditions for sequences of random variables can be found in this paper, where the focus will be on the structural properties of these conditions, and not at all on limit theory.
Abstract: This is a survey of the basic properties of strong mixing conditions for sequences of random variables. The focus will be on the “structural” properties of these conditions, and not at all on limit theory. For a discussion of central limit theorems and related results under these conditions, the reader is referred to Peligrad [60] or Iosifescu [50]. This survey will be divided into eight sections, as follows:
1.
Measures of dependence
2.
Five strong mixing conditions
3.
Mixing conditions for two or more sequences
4.
Mixing conditions for Markov chains
5.
Mixing conditions for Gaussian sequences
6.
Some other special examples
7.
The behavior of the dependence coefficients
8.
Approximation of mixing sequences by other random sequences.
582 citations
528 citations
TL;DR: In this article, it was shown that a Harris-recurrent Markov chain is equivalent to a process having a recurrence point, which leads to simple proofs of the ergodic theorem, existence and uniqueness of stationary measures.
Abstract: Let {X n ; n ≥ 0} be a Harris-recurrent Markov chain on a general state space. It is shown that there is a sequence of random times {N i ; i ≥ 1} such that {X N ; i ≥ 1} are independent and identically distributed. This idea is used to show that {X n } is equivalent to a process having a recurrence point, and to develop a regenerative scheme which leads to simple proofs of the ergodic theorem, existence and uniqueness of stationary measures.
430 citations
Cites background from "The Existence of Stationary Measure..."
...Harris-recurrent chains are known to have unique, a-finite, invariant measures (Harris [6], Orey [8])....
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...A weaker condition, introduced by Harris [6], is the so-called m-recurrence or Harris-recurrence, which requires the existence of a a-finite measure...
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...T. E. Harris, The existence of stationary measures for certain Markov processes, Proc....
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...Under this hypothesis much of the discrete state space theory has been carried over to the general case by Harris [6], Orey [8], and others....
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...A weaker condition, introduced by Harris [6], is the so-called m-recurrence or Harris-recurrence, which requires the existence of a a-finite measure <p on (S, S) such that PX{X„ £ A for some n] = 1 for all A E S with <p(A) > 0....
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References
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TL;DR: In this paper, the authors considered the problem of finite invariant transformations, for which the invariant measure m*(M) of the whole manifold M is finite, and devoted to the characterization of these transformations by their intrinsic properties.
Abstract: holds for any measurable subset a of M, a, being the image of a under T. Such transformations are known to play an important role in dynamics. The motions of a dynamical system, considered in the manifold of states of motion, are equivalent to a one-parameter group of one-to-one transformations. In the case of a conservative system these transformations always possess a positive invariant integral; for instance in the case of a Hamiltonian system the phase volume itself is invariant. The integral m*(a) may-be regarded as another measure on M; thus a transformation of that kind is measure-preserving for a suitably chosen measure. The following paper deals with such transformations, for which the invariant measure m*(M) of the whole manifold M is finite, and is devoted to the characterization of these transformations by their intrinsic properties. Necessary conditions for the existence of a finite invariant measure can be easily derived. For instance no point set a of positive measure can be transformed into a "proper" part of itself, i.e.
48 citations
TL;DR: In this paper, the authors generalize the celebrated ergodic theorem of G. D. Birkhoff to the special case of measure-preserving flows and present an averaging process that can be applied to any flow which is free from dissipative parts.
Abstract: The result presented in this paper is generalization of the celebrated ergodic theorem of G. D. Birkhoff.' Birkhoff's theorem, von Neumann's mean ergodic theorem, and other known results of ergodic theory deal with measure-preserving flows and describe their statistical properties. The condition of measure invrariance seems to play a very essential role in the entire theory, and the question arises whether it is possible to make any general statements about statistical behaviour of flows without assuming the invariance of the underlying measure. We shall answer this question in the affirmative by introducing an averaging process that can be applied to any flow which is free from dissipative parts. In the special case of a measure-preserving flow, our result coincides with Birkhoff's theorem.
47 citations