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{{MSC|60F99}}
 
{{MSC|60F99}}
  
 
[[Category:Limit theorems]]
 
[[Category:Limit theorems]]
  
A property of random variables indicating that their individual contribution as components of a sum is small. This concept is important, for example, in the so-called [[Triangular array|triangular array]]. Let the random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a0137201.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a0137202.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a0137203.png" />) be mutually independent for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a0137204.png" />, and let
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A property of random variables indicating that their individual contribution as components of a sum is small. This concept is important, for example, in the so-called [[Triangular array|triangular array]]. Let the random variables $  X _ {nk }  $(
 +
$  n=1, 2 ,\dots $;  
 +
$  k=1 \dots k _ {n} $)  
 +
be mutually independent for each $  n $,  
 +
and let
  
<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/a/a013/a013720/a0137205.png" /></td> </tr></table>
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$$
 +
S _ {n}  = X _ {n1} + \dots +X _ {n k _ {n}  } .
 +
$$
  
If for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a0137206.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a0137207.png" />, at sufficiently large values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a0137208.png" />, the inequality
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If for all $  \epsilon > 0 $
 +
and  $  \delta > 0 $,  
 +
at sufficiently large values of $  n $,  
 +
the inequality
  
<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/a/a013/a013720/a0137209.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$ \tag{1 }
 +
\max _ {1 \leq  k \leq  k _ {n} } \
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{\mathsf P} ( | X _ {nk} | > \epsilon ) < \delta
 +
$$
  
is satisfied, the individual terms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a01372010.png" /> are called asymptotically negligible (the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a01372011.png" /> then form a so-called zero triangular array). If condition (1) is met, one obtains the following important result: The class of limit distributions for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a01372012.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a01372013.png" /> are certain "centering" constants) coincides with the class of infinitely-divisible distributions (cf. [[Infinitely-divisible distribution|Infinitely-divisible distribution]]). If the distributions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a01372014.png" /> converge to a limit distribution, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a01372015.png" />, and the terms are identically distributed, condition (1) is automatically met. If the requirement for asymptotic negligibility is strengthened by assuming that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a01372016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a01372017.png" /> for all sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a01372018.png" /> one has
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is satisfied, the individual terms $  X _ {nk} $
 
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are called asymptotically negligible (the variables $  X _ {nk }  $
<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/a/a013/a013720/a01372019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
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then form a so-called zero triangular array). If condition (1) is met, one obtains the following important result: The class of limit distributions for $  S _ {n} - A _ {n} $(
 
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$  A _ {n} $
then the following statement is valid: If (2) is met, the limit distribution for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013720/a01372020.png" /> can only be a [[Normal distribution|normal distribution]] (in particular with variance equal to zero, i.e. a degenerate distribution).
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are certain "centering" constants) coincides with the class of infinitely-divisible distributions (cf. [[Infinitely-divisible distribution|Infinitely-divisible distribution]]). If the distributions of $  S _ {n} $
 +
converge to a limit distribution, $  k _ {n} \rightarrow \infty $,  
 +
and the terms are identically distributed, condition (1) is automatically met. If the requirement for asymptotic negligibility is strengthened by assuming that for all $  \epsilon > 0 $
 +
and  $  \delta > 0 $
 +
for all sufficiently large $  n $
 +
one has
  
 +
$$ \tag{2 }
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{\mathsf P} \left ( \max _ {1 \leq  k \leq  k _ {n} } \
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| X _ {nk} | > \epsilon \right )  <  \delta ,
 +
$$
  
 +
then the following statement is valid: If (2) is met, the limit distribution for  $  S _ {n} - A _ {n} $
 +
can only be a [[Normal distribution|normal distribution]] (in particular with variance equal to zero, i.e. a degenerate distribution).
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Feller, "An introduction to probability theory and its applications" , '''2''' , Wiley (1966) pp. 210 {{MR|0210154}} {{ZBL|0138.10207}} </TD></TR></table>
+
{|
 +
|valign="top"|{{Ref|F}}|| W. Feller, [[Feller, "An introduction to probability theory and its  applications"|"An introduction to probability theory and its applications"]], '''2''' , Wiley (1966) pp. 210
 +
|}

Latest revision as of 18:48, 5 April 2020


2020 Mathematics Subject Classification: Primary: 60F99 [MSN][ZBL]

A property of random variables indicating that their individual contribution as components of a sum is small. This concept is important, for example, in the so-called triangular array. Let the random variables $ X _ {nk } $( $ n=1, 2 ,\dots $; $ k=1 \dots k _ {n} $) be mutually independent for each $ n $, and let

$$ S _ {n} = X _ {n1} + \dots +X _ {n k _ {n} } . $$

If for all $ \epsilon > 0 $ and $ \delta > 0 $, at sufficiently large values of $ n $, the inequality

$$ \tag{1 } \max _ {1 \leq k \leq k _ {n} } \ {\mathsf P} ( | X _ {nk} | > \epsilon ) < \delta $$

is satisfied, the individual terms $ X _ {nk} $ are called asymptotically negligible (the variables $ X _ {nk } $ then form a so-called zero triangular array). If condition (1) is met, one obtains the following important result: The class of limit distributions for $ S _ {n} - A _ {n} $( $ A _ {n} $ are certain "centering" constants) coincides with the class of infinitely-divisible distributions (cf. Infinitely-divisible distribution). If the distributions of $ S _ {n} $ converge to a limit distribution, $ k _ {n} \rightarrow \infty $, and the terms are identically distributed, condition (1) is automatically met. If the requirement for asymptotic negligibility is strengthened by assuming that for all $ \epsilon > 0 $ and $ \delta > 0 $ for all sufficiently large $ n $ one has

$$ \tag{2 } {\mathsf P} \left ( \max _ {1 \leq k \leq k _ {n} } \ | X _ {nk} | > \epsilon \right ) < \delta , $$

then the following statement is valid: If (2) is met, the limit distribution for $ S _ {n} - A _ {n} $ can only be a normal distribution (in particular with variance equal to zero, i.e. a degenerate distribution).

Comments

References

[F] W. Feller, "An introduction to probability theory and its applications", 2 , Wiley (1966) pp. 210
How to Cite This Entry:
Asymptotic negligibility. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Asymptotic_negligibility&oldid=23576
This article was adapted from an original article by A.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article