Entropy conditions for subsequences of random variables with applications to empirical processes |
| |
Authors: | István Berkes Walter Philipp Robert Tichy |
| |
Institution: | (1) Technical University Graz, Austria;(2) University of Illinois, Champaign, IL, USA |
| |
Abstract: | We introduce new entropy concepts measuring the size of a given class of increasing sequences of positive integers. Under
the assumption that the entropy function of
is not too large, many strong limit theorems will continue to hold uniformly over all sequences in
. We demonstrate this fact by extending the Chung-Smirnov law of the iterated logarithm on empirical distribution functions
for independent identically distributed random variables as well as for stationary strongly mixing sequences to hold uniformly
over all sequences in
. We prove a similar result for sequences (n
k
ω) mod 1 where the sequence (n
k
) of real numbers satisfies a Hadamard gap condition.
Authors’ addresses: István Berkes, Department of Statistics, Technical University Graz, Steyrergasse 17/IV, A-8010 Graz, Austria;
Walter Philipp, Department of Statistics, University of Illinois, 725 S. Wright Street, Champaign, IL 61820, USA; Robert F.
Tichy, Department of Analysis and Computational Number Theory, Technical University Graz, Steyrergasse 30, A-8010 Graz, Austria |
| |
Keywords: | 2000 Mathematics Subject Classification: 11K38 60F15 |
本文献已被 SpringerLink 等数据库收录! |
|