On the central limit theorem for some birth and death processes

Tymoteusz Chojecki

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica (2011)

  • Volume: 65, Issue: 1
  • ISSN: 0365-1029

Abstract

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Suppose that { X n : n ≥ 0 } is a stationary Markov chain and V is a certain function on a phase space of the chain, called an observable. We say that the observable satisfies the central limit theorem (CLT) if Y n : = N - 1 / 2 ∑ n = 0 N V ( X n ) converge in law to a normal random variable, as N → + ∞ . For a stationary Markov chain with the L 2 spectral gap the theorem holds for all V such that V ( X 0 ) is centered and square integrable, see Gordin [7]. The purpose of this article is to characterize a family of observables V for which the CLT holds for a class of birth and death chains whose dynamics has no spectral gap, so that Gordin’s result cannot be used and the result follows from an application of Kipnis-Varadhan theory.

How to cite

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Tymoteusz Chojecki. "On the central limit theorem for some birth and death processes." Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica 65.1 (2011): null. <http://eudml.org/doc/289828>.

@article{TymoteuszChojecki2011,
abstract = {Suppose that $\lbrace Xn: n \ge 0\rbrace $ is a stationary Markov chain and $V$ is a certain function on a phase space of the chain, called an observable. We say that the observable satisfies the central limit theorem (CLT) if $Y_n :=N^\{-1/2\}\sum _\{n=0\}^N V (X_n)$ converge in law to a normal random variable, as $N \rightarrow +\infty $. For a stationary Markov chain with the $L^2$ spectral gap the theorem holds for all $V$ such that $V (X_0)$ is centered and square integrable, see Gordin [7]. The purpose of this article is to characterize a family of observables $V$ for which the CLT holds for a class of birth and death chains whose dynamics has no spectral gap, so that Gordin’s result cannot be used and the result follows from an application of Kipnis-Varadhan theory.},
author = {Tymoteusz Chojecki},
journal = {Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica},
keywords = {Central limit theorem; Markov chain; Lamperti’s problem; birth and death processes; Kipnis-Varadhan theory; spectral gap},
language = {eng},
number = {1},
pages = {null},
title = {On the central limit theorem for some birth and death processes},
url = {http://eudml.org/doc/289828},
volume = {65},
year = {2011},
}

TY - JOUR
AU - Tymoteusz Chojecki
TI - On the central limit theorem for some birth and death processes
JO - Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
PY - 2011
VL - 65
IS - 1
SP - null
AB - Suppose that $\lbrace Xn: n \ge 0\rbrace $ is a stationary Markov chain and $V$ is a certain function on a phase space of the chain, called an observable. We say that the observable satisfies the central limit theorem (CLT) if $Y_n :=N^{-1/2}\sum _{n=0}^N V (X_n)$ converge in law to a normal random variable, as $N \rightarrow +\infty $. For a stationary Markov chain with the $L^2$ spectral gap the theorem holds for all $V$ such that $V (X_0)$ is centered and square integrable, see Gordin [7]. The purpose of this article is to characterize a family of observables $V$ for which the CLT holds for a class of birth and death chains whose dynamics has no spectral gap, so that Gordin’s result cannot be used and the result follows from an application of Kipnis-Varadhan theory.
LA - eng
KW - Central limit theorem; Markov chain; Lamperti’s problem; birth and death processes; Kipnis-Varadhan theory; spectral gap
UR - http://eudml.org/doc/289828
ER -

References

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  2. Chung, K. L., Markov Chains with Stationary Transition Probabilities, 2nd edition, Springer-Verlag, Berlin, 1967. 
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  5. Durrett, R., Probability Theory and Examples, Wadsworth Publishing Company, Belmont, 1996. 
  6. Feller, W., An Introduction to Probability Theory and its Applications, Vol. II. Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971. 
  7. Gordin, M. I., The central limit theorem for stationary processes, Dokl. Akad. Nauk SSSR 188 (1969), 739-741 (Russian). 
  8. Kipnis, C., Varadhan, S. R. S., Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions, Comm. Math. Phys. 104, no. 1 (1986), 1-19. 
  9. Liggett, T., Stochastic Interacting Systems: Contact, Voter and Exclusion Processes, Grund. der Math. Wissen., 324, Springer-Verlag, Berlin, 1999. 
  10. Menshikov, M.,Wade, A.,Rate of escape and central limit theorem for the supercritical Lamperti problem, Stochastic Process. Appl. 120 (2010), 2078-2099. 
  11. Olla, S., Notes on Central Limits Theorems for Tagged Particles and Diffusions in Random Environment, Etats de la recherche: Milieux Aleatoires CIRM, Luminy, 2000. 

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