IJPAM: Volume 20, No. 2 (2005)
Department of Mathematics
Institute of Mathematics
Maria Curie-Skodowska University
Pl. Marii Curie-Skodowskiej 1, Lublin 20-031, POLAND
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 -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