View source for Attraction, partial domain of

''of an infinitely-divisible distribution''

The set of all distribution functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013930/a0139301.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/a013930/a0139302.png" /> with distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013930/a0139303.png" />, for an appropriate choice of constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013930/a0139304.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013930/a0139305.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013930/a0139306.png" /> and a subsequence of integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013930/a0139307.png" />, the distribution functions of the random variables

<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/a013930/a0139308.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

converge weakly, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013930/a0139309.png" />, to a (given) non-degenerate distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013930/a01393010.png" /> that is infinitely divisible; every infinitely-divisible distribution has a non-empty domain of partial attraction. There exist distribution functions that do not belong to any partial domain of attraction and there also exist distribution functions that belong to the partial domain of attraction of any infinitely-divisible distribution function.

====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></table>



====Comments====
The notion defined in this article is also commonly called the domain of partial attraction.