View source for Attraction domain of a stable distribution

''domain of attraction of a stable distribution''

{{MSC|60F05|60E07}}

[[Category:Limit theorems]]
[[Category:Distribution theory]]

The totality of all distribution functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a0139201.png" /> such that for a sequence of independent identically-distributed random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a0139202.png" /> with distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a0139203.png" /> and for a suitable choice of constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a0139204.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a0139205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a0139206.png" /> the distribution of the random variable

<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/a013920/a0139207.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

converges weakly, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a0139208.png" />, to a non-degenerate distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a0139209.png" />, which is necessarily stable.

One of the fundamental problems in the theory of stable laws is the description of domains of attraction of stable laws. Thus, for the normal distribution, A.Ya. Khinchin, W. Feller and P. Lévy established in 1935 that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392010.png" /> belongs to the domain of attraction of a normal law if and only if, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392011.png" />,

<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/a013920/a01392012.png" /></td> </tr></table>

Later B.V. Gnedenko and W. Doeblin (1940) gave a description of the domain of attraction of a stable law with exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392014.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392015.png" /> belongs to the domain of attraction of a non-degenerate stable law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392016.png" /> with exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392017.png" /> if and only if:

<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/a013920/a01392018.png" /></td> </tr></table>

for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392019.png" />, determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392020.png" />, and

<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/a013920/a01392021.png" /></td> </tr></table>

for each constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392022.png" />. Restriction on the behaviour of the normalizing coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392024.png" /> leads to narrower classes of distribution functions for which the convergence in distribution (*) holds. The set of distribution functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392025.png" /> for which (*) converges weakly, for a suitable choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392029.png" /> to a stable distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392030.png" /> with exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392031.png" />, is called the normal domain of attraction for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392032.png" />. The normal domain of attraction of a normal distribution coincides with the set of non-degenerate distributions with a finite variance.

The normal domain of attraction of a non-degenerate stable distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392033.png" /> with exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392034.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392035.png" />) is formed by the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392036.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/a/a013/a013920/a01392037.png" /></td> </tr></table>

<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/a013920/a01392038.png" /></td> </tr></table>

exist and are finite, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392039.png" /> are determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013920/a01392040.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.V. Gnedenko,   A.N. Kolmogorov,   "Limit distributions for sums of independent random variables" , Addison-Wesley  (1954)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</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><TR><TD valign="top">[3]</TD> <TD valign="top">  V.V. Petrov,   "Sums of independent random variables" , Springer  (1975)  (Translated from Russian)</TD></TR></table>