Conditional zero-one laws
Teoriâ veroâtnostej i ee primeneniâ, Tome 48 (2003) no. 4, pp. 828-834
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We say that a class of events fulfills a
conditional zero-one law if it is a subset of the completion of
the conditioning $\sigma$-algebra. In this case the conditional
probability of an event of the class is an indicator function.
Therefore the conditional probability takes almost surely only
the values zero and one; in the unconditional case the indicator
functions are almost surely constant.
We consider two special zero-one laws. If a
sequence of random variables is conditionally independent, then
its tail $\sigma$-algebra fulfills a conditional zero-one law;
this generalizes Kolmogorov's zero-one law. If the sequence is
even conditionally identically distributed, then its permutable
$\sigma$-algebra, which contains the tail $\sigma$-algebra,
fulfills a conditional zero-one law; this generalizes the
zero-one law of Hewitt and Savage.
Keywords:
conditional probability, conditional independence, zero-one law.
@article{TVP_2003_48_4_a12,
author = {K. Hess},
title = {Conditional zero-one laws},
journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
pages = {828--834},
publisher = {mathdoc},
volume = {48},
number = {4},
year = {2003},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TVP_2003_48_4_a12/}
}
K. Hess. Conditional zero-one laws. Teoriâ veroâtnostej i ee primeneniâ, Tome 48 (2003) no. 4, pp. 828-834. http://geodesic.mathdoc.fr/item/TVP_2003_48_4_a12/