Effective WLLN, SLLN and CLT in statistical models

Ryszard Zieliński¹

¹ Institute of Mathematics Polish Academy of Sciences P.O. Box 21 00-956 Warszawa 10, Poland

Applicationes Mathematicae, Tome 31 (2004) no. 1, pp. 117-125.

Résumé

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.

Détail

DOI : 10.4064/am31-1-10

Keywords: weak laws large numbers wlln strong laws large numbers slln central limit theorems clt statistical models differ those probability theory should uniformly family distributions specified model limit law states every varepsilon there exists inequalities varepsilon satisfied varepsilon explicitly given call law effective trivial obtain effective statistical version wlln bernoulli scheme get slln takes little while clt does uniformly other statistical schemes considered