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An identity in [[Sequential analysis|sequential analysis]] which states that the mathematical expectation of the sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097010/w0970101.png" /> of a random number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097010/w0970102.png" /> of independent, identically-distributed random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097010/w0970103.png" /> is equal to the product of the mathematical expectations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097010/w0970104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097010/w0970105.png" />:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097010/w0970106.png" /></td> </tr></table>
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A sufficient condition for the Wald identity to be valid is that the mathematical expectations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097010/w0970107.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097010/w0970108.png" /> in fact exist, and for the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097010/w0970109.png" /> to be a Markov time (i.e. for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097010/w09701010.png" /> the event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097010/w09701011.png" /> is determined by the values of the random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097010/w09701012.png" /> or, which is the same thing, the event <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097010/w09701013.png" /> belongs to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097010/w09701014.png" />-algebra generated by the random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097010/w09701015.png" />). Wald's identity is a particular case of a fundamental theorem in sequential analysis stating that
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An identity in [[Sequential analysis|sequential analysis]] which states that the mathematical expectation of the sum  $  S _  \tau  = X _ {1} + \dots + X _  \tau  $
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of a random number  $  \tau $
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of independent, identically-distributed random variables $  X _ {1} , X _ {2} \dots $
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is equal to the product of the mathematical expectations  $  {\mathsf E}  X _ {1} $
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and  $  {\mathsf E}  \tau $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097010/w09701016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$
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{\mathsf E} ( X _ {1} + \dots + X _  \tau  )  = \
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{\mathsf E} X _ {1} \cdot {\mathsf E} \tau .
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$$
  
for all complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097010/w09701017.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097010/w09701018.png" /> exists and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w097/w097010/w09701019.png" />. It was established by A. Wald [[#References|[1]]].
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A sufficient condition for the Wald identity to be valid is that the mathematical expectations  $  {\mathsf E} | X _ {1} | $
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and  $  {\mathsf E} \tau $
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in fact exist, and for the random variable  $  \tau $
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to be a Markov time (i.e. for any  $  n = 1, 2 \dots $
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the event  $  \{ \tau = n \} $
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is determined by the values of the random variables  $  X _ {1} \dots X _ {n} $
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or, which is the same thing, the event  $  \{ \tau = n \} $
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belongs to the  $  \sigma $-
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algebra generated by the random variables  $  X _ {1} \dots X _ {n} $).  
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Wald's identity is a particular case of a fundamental theorem in sequential analysis stating that
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$$ \tag{* }
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{\mathsf E} \left [ e ^ {\lambda S _  \tau  } ( \phi ( \lambda )) ^ {- \tau }
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\right ]  =  1
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$$
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for all complex  $  \lambda $
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for which $  \phi ( \lambda ) = {\mathsf E} e ^ {\lambda X _ {1} } $
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exists and $  | \phi ( \lambda ) | \geq  1 $.  
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It was established by A. Wald [[#References|[1]]].
  
 
====References====
 
====References====

Latest revision as of 08:28, 6 June 2020


An identity in sequential analysis which states that the mathematical expectation of the sum $ S _ \tau = X _ {1} + \dots + X _ \tau $ of a random number $ \tau $ of independent, identically-distributed random variables $ X _ {1} , X _ {2} \dots $ is equal to the product of the mathematical expectations $ {\mathsf E} X _ {1} $ and $ {\mathsf E} \tau $:

$$ {\mathsf E} ( X _ {1} + \dots + X _ \tau ) = \ {\mathsf E} X _ {1} \cdot {\mathsf E} \tau . $$

A sufficient condition for the Wald identity to be valid is that the mathematical expectations $ {\mathsf E} | X _ {1} | $ and $ {\mathsf E} \tau $ in fact exist, and for the random variable $ \tau $ to be a Markov time (i.e. for any $ n = 1, 2 \dots $ the event $ \{ \tau = n \} $ is determined by the values of the random variables $ X _ {1} \dots X _ {n} $ or, which is the same thing, the event $ \{ \tau = n \} $ belongs to the $ \sigma $- algebra generated by the random variables $ X _ {1} \dots X _ {n} $). Wald's identity is a particular case of a fundamental theorem in sequential analysis stating that

$$ \tag{* } {\mathsf E} \left [ e ^ {\lambda S _ \tau } ( \phi ( \lambda )) ^ {- \tau } \right ] = 1 $$

for all complex $ \lambda $ for which $ \phi ( \lambda ) = {\mathsf E} e ^ {\lambda X _ {1} } $ exists and $ | \phi ( \lambda ) | \geq 1 $. It was established by A. Wald [1].

References

[1] A. Wald, "Sequential analysis", Wiley (1952)
[2] W. Feller, "An introduction to probability theory and its applications", 1, Wiley (1957) pp. Chapt.14

Comments

The general result (*) is (also) referred to as Wald's formula.

References

[a1] A.V. [A.V. Skorokhod] Skorohod, "Random processes with independent increments" , Kluwer (1991) pp. 23 (Translated from Russian)
How to Cite This Entry:
Wald identity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wald_identity&oldid=49166
This article was adapted from an original article by A.N. Shiryaev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article