View source for Absolutely-unbiased sequence

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its  applications"]], '''2''', Wiley (1966) pp. 210</TD></TR></table>