# Absolutely-unbiased sequence

A sequence of random variables \$ X _ {1} \dots X _ {n} \$ for which the conditions

\$\$ {\mathsf E} ( X _ {1} ) = 0 \ \textrm{ and } \ {\mathsf E} ( X _ {n+1} \mid X _ {1} \dots X _ {n} ) = 0 \$\$

are fulfilled, for \$ n = 1, 2 ,\dots \$. The partial sums \$ S _ {n} = X _ {1} + \dots + X _ {n} \$ of an absolutely-unbiased sequence form a martingale. These two types of sequences are interconnected as follows: The sequence \$ \{ Y _ {n} \} \$ forms a martingale if and only if it is of the form \$ Y _ {n} = X _ {1} + \dots + X _ {n} + c \$( \$ n = 1, 2 \dots \$ and \$ c = {\mathsf E} ( Y _ {1} ) \$ is a constant), where \$ \{ X _ {n} \} \$ is an absolutely-unbiased sequence. Thus, all martingales are related to partial sums of certain absolutely-unbiased sequences. Simple examples of absolutely-unbiased sequences are sequences of independent random variables with mathematical expectation zero. Besides the term "unbiased" the term "fair" — with the related concept of a "fair play" , is also employed.

In [a1] the term "absolutely fair sequenceabsolutely fair" is used instead of absolutely-unbiased.

#### References

 [a1] W. Feller, "An introduction to probability theory and its applications", 2, Wiley (1966) pp. 210
How to Cite This Entry:
Absolutely-unbiased sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Absolutely-unbiased_sequence&oldid=45004
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article