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Generalization To The Non-Ergodic Case

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1766)

Abstract

Let (X n)n be a Markov chain on a measurable space E with a transition kernel Q, and let β be a Banach space composed of measurable functions on E such that Q is quasi-compact on β, that supn>1 ||Q n|| < +∞, and that Ker(Q - 1) ∩ β admits a basis of positive functions. Let ξ : E → ℝ be a measurable function. Under standard assumptions on the Fourier kernels Q(t)(x, dy) = eitξ(y) Q(x, dy), we prove central limit theorems, local theorems and renewal theorems for (ξ(X n))n.

Keywords

  • Probability Measure
  • Central Limit Theorem
  • Banach Lattice
  • Local Theorem
  • Markov Kernel

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© 2001 Springer-Verlag Berlin Heidelberg

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Herve, L. (2001). Generalization To The Non-Ergodic Case. In: Hennion, H., Hervé, L. (eds) Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness. Lecture Notes in Mathematics, vol 1766. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44623-0_15

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  • DOI: https://doi.org/10.1007/3-540-44623-0_15

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42415-4

  • Online ISBN: 978-3-540-44623-1

  • eBook Packages: Springer Book Archive