View source for Zone of normal attraction

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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|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====
<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====
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]]),  $  \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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.V. Petrov,   "Sums of independent random variables" , Springer  (1975)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R.J. Serfling,   "Approximation theorems of mathematical statistics" , Wiley  (1980)  pp. 6, 96</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.D. [A.D. Ventsel'] Wentzell,   "Limit theorems on large deviations for Markov stochastic processes" , Kluwer  (1990)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L. Saulis,   V.A. Statulevicius,   "Limit theorems for large deviations" , Kluwer  (1991)  (Translated from Russian)</TD></TR></table>