It is easy to notice that no sequence of estimators of the probability of success θ in a Bernoulli scheme can converge (when standardized) to N(0,1) uniformly in θ ∈ ]0,1[. We show that the uniform asymptotic normality can be achieved if we allow the sample size, that is, the number of Bernoulli trials, to be chosen sequentially.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-am34-2-6,
author = {Wojciech Niemiro and Ryszard Zieli\'nski},
title = {Uniform asymptotic normality for the Bernoulli scheme},
journal = {Applicationes Mathematicae},
volume = {34},
year = {2007},
pages = {215-221},
zbl = {1121.60019},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-am34-2-6}
}
Wojciech Niemiro; Ryszard Zieliński. Uniform asymptotic normality for the Bernoulli scheme. Applicationes Mathematicae, Tome 34 (2007) pp. 215-221. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-am34-2-6/