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Journal ArticleDOI

Central limit theorem for an additive functional of a Markov process, stable in the Wesserstein metric

01 Jan 2008-Annales Umcs, Mathematica (De Gruyter Open)-Vol. 62, Iss: 1, pp 149-159
TL;DR: In this article, the authors studied the question of the law of large numbers and central limit theorem for an additive functional of a Markov process taking values in a Polish space that has Feller property under the assumption that the process is asymptotically contractive in the Wasserstein metric.
Abstract: We study the question of the law of large numbers and central limit theorem for an additive functional of a Markov processes taking values in a Polish space that has Feller property under the assumption that the process is asymptotically contractive in the Wasserstein metric.
Citations
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Journal ArticleDOI
TL;DR: In this article, the authors prove the central limit theorem for an additive functional of the form ∫ 0 T ψ (X s ) d s, provided that the dual transition probability semigroup, defined on measures, is strongly contractive in an appropriate Wasserstein metric.

66 citations

Journal ArticleDOI
TL;DR: In this paper, the Central Limit Theorem for random dynamical systems with randomly chosen jumps is established and the choice of deterministic dynamical system and jumps depends on a position.
Abstract: We consider random dynamical systems with randomly chosen jumps. The choice of deterministic dynamical system and jumps depends on a position. The Central Limit Theorem for random dynamical systems is established.

9 citations

Journal ArticleDOI
TL;DR: In this paper, the authors consider random dynamical systems with randomly chosen jumps and prove the existence of an exponentially attractive invariant measure and the strong law of large numbers for such systems.
Abstract: We consider random dynamical systems with randomly chosen jumps. The choice of deterministic dynamical system and jumps depends on a position. We prove the existence of an exponentially attractive invariant measure and the strong law of large numbers.

9 citations

Posted Content
TL;DR: In this paper, the existence of an exponentially attractive invariant measure and the strong law of large numbers was proved for random dynamical systems with randomly chosen jumps. But the choice of deterministic dynamical system and jumps depends on a position.
Abstract: We cosider random dynamical systems with randomly chosen jumps. The choice of deterministic dynamical system and jumps depends on a position. We proove the existence of an exponentially attractive invariant measure and the strong law of large numbers.

8 citations

Journal ArticleDOI
TL;DR: In this article, the strong law of large numbers, central limit theorem, and the law of iterated logarithm are established for additive functionals of path-dependent stochastic differential equations.
Abstract: By using limit theorems of uniform mixing Markov processes and martingale difference sequences, the strong law of large numbers, central limit theorem, and the law of iterated logarithm are established for additive functionals of path-dependent stochastic differential equations.

7 citations

References
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Book
01 Jan 1968
TL;DR: Weak Convergence in Metric Spaces as discussed by the authors is one of the most common modes of convergence in metric spaces, and it can be seen as a form of weak convergence in metric space.
Abstract: Weak Convergence in Metric Spaces. The Space C. The Space D. Dependent Variables. Other Modes of Convergence. Appendix. Some Notes on the Problems. Bibliographical Notes. Bibliography. Index.

13,153 citations

Journal ArticleDOI
TL;DR: In this paper, a functional central limit theorem for additive functionals of stationary reversible ergodic Markov chains was proved under virtually no assumptions other than the necessary ones, and they used these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk.
Abstract: We prove a functional central limit theorem for additive functionals of stationary reversible ergodic Markov chains under virtually no assumptions other than the necessary ones. We use these results to study the asymptotic behavior of a tagged particle in an infinite particle system performing simple excluded random walk.

909 citations


"Central limit theorem for an additi..." refers background or methods in this paper

  • ...It is based on the martingale approach of Kipnis and Varadhan, see [6]....

    [...]

  • ...[6, 9, 8], it is usually assumed that the process under consideration is stationary and its equilibrium state μ∗ is stable in some sense, usually in the L2, or total variation norm....

    [...]

Book
02 Mar 1989
TL;DR: This book discusses set theory, vector spaces, and Taylor's theorem with remainder, as well as general topology, measurement, and differentiation, and introduces probability theory.
Abstract: This classic textbook offers a clear exposition of modern probability theory and of the interplay between the properties of metric spaces and probability measures. The first half of the book gives an exposition of real analysis: basic set theory, general topology, measure theory, integration, an introduction to functional analysis in Banach and Hilbert spaces, convex sets and functions and measure on topological spaces. The second half introduces probability based on measure theory, including laws of large numbers, ergodic theorems, the central limit theorem, conditional expectations and martingale's convergence. A chapter on stochastic processes introduces Brownian motion and the Brownian bridge. The edition has been made even more self-contained than before; it now includes a foundation of the real number system and the Stone-Weierstrass theorem on uniform approximation in algebras of functions. Several other sections have been revised and improved, and the comprehensive historical notes have been further amplified. A number of new exercises have been added, together with hints for solution.

786 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that there is a constant R \vartheta, where R \leq \rho/(\rho - \varta) = \frac{1}{(1 - \lambda)^2}
Abstract: Recent results for geometrically ergodic Markov chains show that there exist constants $R \vartheta$, where $\lbrack 1 - \vartheta\rbrack^{-1} = \frac{1}{(1 - \lambda)^2} \lbrack 1 - \lambda + b + b^2 + \zeta_\alpha(b(1 - \lambda) + b^2)\rbrack$ and $\zeta_\alpha \leq \big(\frac{32 - 8 \delta^2}{\delta^3}\big) \big(\frac{b}{1 - \lambda}\big)^2,$ and we can then choose $R \leq \rho/(\rho - \vartheta)$. The bounds for general small sets $C$ are similar but more complex. We apply these to simple queuing models and Markov chain Monte Carlo algorithms, although in the latter the bounds are clearly too large for practical application in the case considered.

319 citations


"Central limit theorem for an additi..." refers background in this paper

  • ...150 A. Walczuk of a Markov process is one of the most fundamental in probability theory and there exists a rich literature on the subject, see e.g. the monograph of Meyn and Tweedie [7] and the citations therein....

    [...]

  • ...[7] Meyn, S. P., Tweedie, R. L., Computable bounds for geometric convergence rates of Markov chains, Ann....

    [...]

  • ...the monograph of Meyn and Tweedie [7] and the citations therein....

    [...]

01 Jan 1982
TL;DR: In this paper, a survey of the central limit theorems for discrete time martingales with continuous time is presented, and several related sets of conditions for convergence are formulated, where conditions are given in terms of conditional moments of truncated variables.
Abstract: This survey paper consists of two parts. In the first part (up to and including setion 3) we review the central limit theorems for discrete time martingales, and show that many different sets of conditions for convergence may be reduced to one basic set, where conditions are given in terms of conditional moments of truncated variables, given the past. In the second part (sections 4 and 5) we first recall some basic facts from the modern "French" theory of stochastic processes, then show that Rebolledo's recent functional limit theorems for martingales with continuous time can be deduced from the limit theorems for discrete time martingales. Again, several related sets of conditions for convergence are formulated.

229 citations