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''Kolmogorov three-series theorem, three-series criterion''
 
''Kolmogorov three-series theorem, three-series criterion''
  
For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092740/t0927401.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092740/t0927402.png" /> be the truncation function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092740/t0927403.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092740/t0927404.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092740/t0927405.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092740/t0927406.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092740/t0927407.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092740/t0927408.png" />.
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For each $  s> 0 $,  
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let $  \tau _ {s} $
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be the truncation function $  \tau _ {s} ( x)= s $
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for $  x \geq  s $,  
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$  \tau _ {s} ( x) = x $
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for $  | x | \leq  s $,  
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$  \tau _ {s} ( x)= - s $
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for $  x \leq  - s $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092740/t0927409.png" /> be independent random variables with distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092740/t09274010.png" />. Consider the sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092740/t09274011.png" />, with distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092740/t09274012.png" />. In order that these convolutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092740/t09274013.png" /> tend to a proper limit distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092740/t09274014.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092740/t09274015.png" />, it is necessary and sufficient that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092740/t09274016.png" />,
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Let $  X _ {1} , X _ {2} \dots $
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be independent random variables with distributions $  F _ {1} , F _ {2} ,\dots $.  
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Consider the sums $  S _ {n} = X _ {1} + \dots + X _ {n} $,  
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with distributions $  F _ {1} \star \dots \star F _ {n} $.  
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In order that these convolutions $  F _ {1} \star \dots \star F _ {n} $
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tend to a proper limit distribution $  F $
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as $  n \rightarrow \infty $,  
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it is necessary and sufficient that for all $  s> 0 $,
  
<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/t/t092/t092740/t09274017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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$$ \tag{a1 }
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\sum _ { k } {\mathsf P} \{ | X _ {k} | > s \}  < \infty ,
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$$
  
<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/t/t092/t092740/t09274018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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$$ \tag{a2 }
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\sum  \mathop{\rm Var} ( X _ {k} ^ { \prime } ) < \infty ,
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$$
  
<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/t/t092/t092740/t09274019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
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$$ \tag{a3 }
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\sum _ { k= 1} ^ { n }  {\mathsf E} ( X _ {k} ^ { \prime } ) \rightarrow  m ,
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$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092740/t09274020.png" />.
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where $  X _ {k} ^ { \prime } = \tau _ {s} ( X _ {k} ) $.
  
This can be reformulated as the Kolmogorov three-series theorem: The series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092740/t09274021.png" /> converges with probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092740/t09274022.png" /> if (a1)–(a3) hold, and it converges with probability zero otherwise.
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This can be reformulated as the Kolmogorov three-series theorem: The series $  \sum X _ {k} $
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converges with probability $  1 $
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if (a1)–(a3) hold, and it converges with probability zero otherwise.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Loève, "Probability theory", Princeton Univ. Press (1963) pp. Sect. 16.3</TD></TR><TR><TD valign="top">[a2]</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 (1971) pp. Sect. IX.9</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Loève, "Probability theory", Princeton Univ. Press (1963) pp. Sect. 16.3</TD></TR><TR><TD valign="top">[a2]</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 (1971) pp. Sect. IX.9</TD></TR></table>

Latest revision as of 16:15, 13 January 2021


Kolmogorov three-series theorem, three-series criterion

For each $ s> 0 $, let $ \tau _ {s} $ be the truncation function $ \tau _ {s} ( x)= s $ for $ x \geq s $, $ \tau _ {s} ( x) = x $ for $ | x | \leq s $, $ \tau _ {s} ( x)= - s $ for $ x \leq - s $.

Let $ X _ {1} , X _ {2} \dots $ be independent random variables with distributions $ F _ {1} , F _ {2} ,\dots $. Consider the sums $ S _ {n} = X _ {1} + \dots + X _ {n} $, with distributions $ F _ {1} \star \dots \star F _ {n} $. In order that these convolutions $ F _ {1} \star \dots \star F _ {n} $ tend to a proper limit distribution $ F $ as $ n \rightarrow \infty $, it is necessary and sufficient that for all $ s> 0 $,

$$ \tag{a1 } \sum _ { k } {\mathsf P} \{ | X _ {k} | > s \} < \infty , $$

$$ \tag{a2 } \sum \mathop{\rm Var} ( X _ {k} ^ { \prime } ) < \infty , $$

$$ \tag{a3 } \sum _ { k= 1} ^ { n } {\mathsf E} ( X _ {k} ^ { \prime } ) \rightarrow m , $$

where $ X _ {k} ^ { \prime } = \tau _ {s} ( X _ {k} ) $.

This can be reformulated as the Kolmogorov three-series theorem: The series $ \sum X _ {k} $ converges with probability $ 1 $ if (a1)–(a3) hold, and it converges with probability zero otherwise.

References

[a1] M. Loève, "Probability theory", Princeton Univ. Press (1963) pp. Sect. 16.3
[a2] W. Feller, "An introduction to probability theory and its applications", 2, Wiley (1971) pp. Sect. IX.9
How to Cite This Entry:
Three-series theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Three-series_theorem&oldid=25947