# On the central limit theorem for some birth and death processes

• Volume: 65, Issue: 1, page 21-31
• ISSN: 2083-7402

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## Abstract

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Suppose that {Xn, 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 [...] [...] converge in law to a normal random variable, as N → +∞. For a stationary Markov chain with the L2 spectral gap the theorem holds for all V such that V (X0) 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 UMCS, Mathematica 65.1 (2011): 21-31. <http://eudml.org/doc/267629>.

@article{TymoteuszChojecki2011,
abstract = {Suppose that \{Xn, 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 [...] [...] converge in law to a normal random variable, as N → +∞. For a stationary Markov chain with the L2 spectral gap the theorem holds for all V such that V (X0) 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 UMCS, Mathematica},
keywords = {Central limit theorem; Markov chain; Lamperti's problem; birth and death processes; Kipnis-Varadhan theory; spectral gap; central limit theorem},
language = {eng},
number = {1},
pages = {21-31},
title = {On the central limit theorem for some birth and death processes},
url = {http://eudml.org/doc/267629},
volume = {65},
year = {2011},
}

TY - JOUR
AU - Tymoteusz Chojecki
TI - On the central limit theorem for some birth and death processes
JO - Annales UMCS, Mathematica
PY - 2011
VL - 65
IS - 1
SP - 21
EP - 31
AB - Suppose that {Xn, 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 [...] [...] converge in law to a normal random variable, as N → +∞. For a stationary Markov chain with the L2 spectral gap the theorem holds for all V such that V (X0) 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; central limit theorem
UR - http://eudml.org/doc/267629
ER -

## References

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5. Durrett, R., Probability Theory and Examples, Wadsworth Publishing Company, Belmont, 1996. Zbl1202.60001
6. Feller, W., An Introduction to Probability Theory and its Applications, Vol. II. Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971. Zbl0219.60003
7. Gordin, M. I., The central limit theorem for stationary processes, Dokl. Akad. Nauk SSSR 188 (1969), 739-741 (Russian). Zbl0212.50005
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. Zbl0588.60058
9. Liggett, T., Stochastic Interacting Systems: Contact, Voter and Exclusion Processes, Grund. der Math. Wissen., 324, Springer-Verlag, Berlin, 1999. Zbl0949.60006
10. Menshikov, M.,Wade, A., Rate of escape and central limit theorem for the supercritical Lamperti problem, Stochastic Process. Appl. 120 (2010), 2078-2099.[WoS] Zbl1207.60032
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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