General Glivenko-Cantelli theorems

A Glivenko–Cantelli theorem is a fundamental result in statistics. It says that an empirical distribution function uniformly approximates the true distribution function for a sufficiently large sample size. We prove general Glivenko–Cantelli theorems for three types of sequences of random variables:...

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Bibliographic Details
Published in:Stat (International Statistical Institute) 2016, Vol.5 (1), p.306-311
Main Authors: Athreya, Krishna B., Roy, Vivekananda
Format: Article
Language:English
Subjects:
Convergence
convergence in distribution
Distribution functions
empirical distribution
Glivenko-Cantelli
Harris recurrence
Markov analysis
Markov chains
Monte Carlo methods
Monte Carlo simulation
Random variables
regenerative sequence
Samples
Statistical analysis
Statistics
Theorems
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Summary:A Glivenko–Cantelli theorem is a fundamental result in statistics. It says that an empirical distribution function uniformly approximates the true distribution function for a sufficiently large sample size. We prove general Glivenko–Cantelli theorems for three types of sequences of random variables: delayed regenerative, delayed stationary and delayed exchangeable. In particular, our results hold for irreducible Harris recurrent Markov chains that admit a stationary probability distribution but are not necessarily in the stationary state. We also do not assume any mixing conditions on the Markov chain. This is useful in the application of Markov chain Monte Carlo methods. A key tool used is a generalized version of Polya's theorem on the convergence of distribution functions. Copyright © 2016 John Wiley & Sons, Ltd.
ISSN:2049-1573
2049-1573
DOI:10.1002/sta4.128