IJPAM: Volume 20, No. 2 (2005)


Dariusz Majerek$^1$, Wioletta Nowak$^2$, Wies\law Zieba$^3$
$^{1,2}$Department of Mathematics
Technical University
Lublin, POLAND
$^1$e-mail: majerek@antenor.pol.lublin.pl
$^2$e-mail: wnowak@antenor.pol.lublin.pl
$^3$Institute of Mathematics
Maria Curie-Sk\lodowska University
Pl. Marii Curie-Sk\lodowskiej 1, Lublin 20-031, POLAND
e-mail: zieba@golem.umcs.lublin.pl

Abstract.The aim of this note is to give a conditional version of Kolmogorov's strong law of large numbers. A strong law of large numbers was generalized in many ways. One of the assumptions, which was weakened, was the independence condition (for example for martingales increments).

In this paper we consider sequences of $\mathcal{F}$-independence of random variables. Note that conditional independence does not imply independence, the opposite implication is also not true, as incorrectly given in the book [#!Jordan!#]. In the second part of this paper we prove a conditional version of the Kolmogorov's strong law of large numbers.

Received: October 25, 2004

AMS Subject Classification: 60F15

Key Words and Phrases: independence, conditional expectation, law of large numbers

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 20
Issue: 2