Difference between revisions of "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==== | ====Comments==== | ||
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====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <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> |
Latest revision as of 16:08, 1 April 2020
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 |
Absolutely-unbiased sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Absolutely-unbiased_sequence&oldid=13957