Nonstandard central limit theorems for Markov chains

Bernardo B. De Andrade; Charles J. Geyer

doi:10.1142/S021848851100699X

Nonstandard central limit theorems for Markov chains

Bernardo B. De Andrade, Charles J. Geyer

Statistics (Twin Cities)

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

We prove a nonstandard central limit theorem for strongly mixing stationary sequences using the "radically elementary" probability theory developed by Nelson (1987) and use this result to obtain a nonstandard Markov chain central limit theorem. An analogue of a Gaussian autoregressive process is used to illustrate how to obtain the rate of convergence. A Markov chain nonstandard functional central limit theorem is also proved under conditions similar to square integrability of the underlying martingale.

Original language	English (US)
Pages (from-to)	251-274
Number of pages	24
Journal	International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems
Volume	19
Issue number	2
DOIs	https://doi.org/10.1142/S021848851100699X
State	Published - Apr 2011

Bibliographical note

Funding Information:
The first author would like to thank FINATEC/UNB (Grant Fomento-4/09) for the partial support given during his stay at the Department of Economics, University of Brasília, while this work was being completed.

Keywords

Central limit theorem
Markov chains
functional CLT
nonstandard analysis

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10.1142/S021848851100699X

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AU - Geyer, Charles J.

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N2 - We prove a nonstandard central limit theorem for strongly mixing stationary sequences using the "radically elementary" probability theory developed by Nelson (1987) and use this result to obtain a nonstandard Markov chain central limit theorem. An analogue of a Gaussian autoregressive process is used to illustrate how to obtain the rate of convergence. A Markov chain nonstandard functional central limit theorem is also proved under conditions similar to square integrability of the underlying martingale.

AB - We prove a nonstandard central limit theorem for strongly mixing stationary sequences using the "radically elementary" probability theory developed by Nelson (1987) and use this result to obtain a nonstandard Markov chain central limit theorem. An analogue of a Gaussian autoregressive process is used to illustrate how to obtain the rate of convergence. A Markov chain nonstandard functional central limit theorem is also proved under conditions similar to square integrability of the underlying martingale.

KW - Central limit theorem

KW - Markov chains

KW - functional CLT

KW - nonstandard analysis

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ER -

Nonstandard central limit theorems for Markov chains

Abstract

Bibliographical note

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