Effective WLLN, SLLN and CLT in statistical models

Volume 31 / 2004

Ryszard Zieliński Applicationes Mathematicae 31 (2004), 117-125 MSC: 62E12, 62E20, 60F05, 60F15. DOI: 10.4064/am31-1-10

Abstract

Weak laws of large numbers (WLLN), strong laws of large numbers (SLLN), and central limit theorems (CLT) in statistical models differ from those in probability theory in that they should hold uniformly in the family of distributions specified by the model. If a limit law states that for every $\varepsilon >0$ there exists $N$ such that for all $n>N$ the inequalities $|\xi _n|<\varepsilon $ are satisfied and $N=N(\varepsilon )$ is explicitly given then we call the law effective. It is trivial to obtain an effective statistical version of WLLN in the Bernoulli scheme, to get SLLN takes a little while, but CLT does not hold uniformly. Other statistical schemes are also considered.

Authors

  • Ryszard ZielińskiInstitute of Mathematics
    Polish Academy of Sciences
    P.O. Box 21
    00-956 Warszawa 10, Poland
    e-mail

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