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Difference between revisions of "Zone of normal attraction"

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$$  
 
$$  
Z _ {n} =   
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Z _ {n} =  \frac{1}{\sigma \sqrt n } \sum_{j=1} ^ { n }  ( X _ {j} - a ) ,
\frac{1}{\sigma \sqrt n }
 
\sum _ { j= } 1 ^ { n }  ( X _ {j} - a ) ,
 
 
$$
 
$$
  
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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.A. Ibragimov,  Yu.V. Linnik,  "Independent and stationary sequences of random variables" , Wolters-Noordhoff  (1971)  (Translated from Russian)</TD></TR></table>
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<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  I.A. Ibragimov,  Yu.V. Linnik,  "Independent and stationary sequences of random variables" , Wolters-Noordhoff  (1971)  (Translated from Russian)</TD></TR>
 +
</table>
  
 
====Comments====
 
====Comments====
 
A rather general formulation of the problems of large deviations is as follows, [[#References|[a3]]]. Suppose that for a family of stochastic processes  $  \xi  ^ {a} ( t) $
 
A rather general formulation of the problems of large deviations is as follows, [[#References|[a3]]]. Suppose that for a family of stochastic processes  $  \xi  ^ {a} ( t) $
a result of the law-of-large-numbers type holds (cf. [[Law of large numbers|Law of large numbers]]),  $  \xi  ^ {a} \rightarrow x $
+
a result of the law-of-large-numbers type holds (cf. [[Law of large numbers]]),  $  \xi  ^ {a} \rightarrow x $
 
as  $  a \rightarrow \infty $.  
 
as  $  a \rightarrow \infty $.  
 
Problems on large deviations of the process  $  \xi  ^ {a} ( t) $
 
Problems on large deviations of the process  $  \xi  ^ {a} ( t) $

Latest revision as of 20:13, 10 January 2024


A domain of the form $ 0 \leq x \leq \psi ( n) $ for which

$$ \frac{ {\mathsf P} \{ Z _ {n} \geq x \} }{ {\mathsf P} \{ Y \geq x \} } \rightarrow 1 \ \textrm{ or } \ \ \frac{ {\mathsf P} \{ Z _ {n} \leq - x \} }{ {\mathsf P} \{ Y \leq - x \} } \rightarrow 1 $$

as $ n \rightarrow \infty $, where $ \{ \psi ( n) \} \uparrow \infty $, $ \{ Z _ {n} \} $ is a sequence of random variables and $ Y $ is a random variable with a normal distribution. Zones of normal attraction have been studied for the case

$$ Z _ {n} = \frac{1}{\sigma \sqrt n } \sum_{j=1} ^ { n } ( X _ {j} - a ) , $$

where $ \{ X _ {n} \} $ is a sequence of independent, identically-distributed random variables with mathematical expectation $ a $ and finite positive variance $ \sigma ^ {2} $.

References

[1] I.A. Ibragimov, Yu.V. Linnik, "Independent and stationary sequences of random variables" , Wolters-Noordhoff (1971) (Translated from Russian)

Comments

A rather general formulation of the problems of large deviations is as follows, [a3]. Suppose that for a family of stochastic processes $ \xi ^ {a} ( t) $ a result of the law-of-large-numbers type holds (cf. Law of large numbers), $ \xi ^ {a} \rightarrow x $ as $ a \rightarrow \infty $. Problems on large deviations of the process $ \xi ^ {a} ( t) $ from its most probable path $ x( t) $ for large values of $ a $ are concerned with the limiting behaviour as $ a \rightarrow \infty $ of the infinitesimal probabilities $ p ^ {a} ( \xi ^ {a} \in A ) $ for measurable sets $ A $ that are at a positive distance from the non-random limiting function $ x $( in a suitable function space (space of paths)). Problems concerning the asymptotics as $ a \rightarrow \infty $ of expectations of the form $ {\mathsf E} ^ {a} [ f ^ { a } ( \xi ^ {a} )] $ also form part of large deviation theory if the main part of these expectations for large values of $ a $ comes from the low probability values of $ \xi ^ {a} $.

References

[a1] V.V. Petrov, "Sums of independent random variables" , Springer (1975) (Translated from Russian)
[a2] R.J. Serfling, "Approximation theorems of mathematical statistics" , Wiley (1980) pp. 6, 96
[a3] A.D. [A.D. Ventsel'] Wentzell, "Limit theorems on large deviations for Markov stochastic processes" , Kluwer (1990) (Translated from Russian)
[a4] L. Saulis, V.A. Statulevicius, "Limit theorems for large deviations" , Kluwer (1991) (Translated from Russian)
How to Cite This Entry:
Zone of normal attraction. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zone_of_normal_attraction&oldid=54965
This article was adapted from an original article by V.V. Petrov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article