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.

Comments

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