View source for Zone of normal attraction

A domain of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099310/z0993101.png" /> for which

<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/z/z099/z099310/z0993102.png" /></td> </tr></table>

as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099310/z0993103.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099310/z0993104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099310/z0993105.png" /> is a sequence of random variables and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099310/z0993106.png" /> is a random variable with a [[Normal distribution|normal distribution]]. Zones of normal attraction have been studied for the case

<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/z/z099/z099310/z0993107.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099310/z0993108.png" /> is a sequence of independent, identically-distributed random variables with mathematical expectation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099310/z0993109.png" /> and finite positive variance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099310/z09931010.png" />.

====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099310/z09931011.png" /> a result of the law-of-large-numbers type holds (cf. [[Law of large numbers|Law of large numbers]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099310/z09931012.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099310/z09931013.png" />. Problems on large deviations of the process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099310/z09931014.png" /> from its most probable path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099310/z09931015.png" /> for large values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099310/z09931016.png" /> are concerned with the limiting behaviour as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099310/z09931017.png" /> of the infinitesimal probabilities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099310/z09931018.png" /> for measurable sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099310/z09931019.png" /> that are at a positive distance from the non-random limiting function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099310/z09931020.png" /> (in a suitable function space (space of paths)). Problems concerning the asymptotics as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099310/z09931021.png" /> of expectations of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099310/z09931022.png" /> also form part of large deviation theory if the main part of these expectations for large values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099310/z09931023.png" /> comes from the low probability values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099310/z09931024.png" />.

====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>