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Attraction, partial domain of

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of an infinitely-divisible distribution

The set of all distribution functions $F(x)$ such that for a sequence of independent identically-distributed random variables $X_1,X_2,\ldots$ with distribution function $F$, for an appropriate choice of constants $A_n$ and $B_n>0$, $n=1,2,\ldots$ and a subsequence of integers $n_1 < n_2 < \cdots$, the distribution functions of the random variables $$ \frac{ \sum_{i=1}^{n_k} X_i - A_{n_k} }{ B_{n_k} } $$ converge weakly, as $k\rightarrow\infty$, to a (given) non-degenerate distribution function $V(x)$ 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

[1] B.V. Gnedenko, A.N. Kolmogorov, "Limit distributions for sums of independent random variables" , Addison-Wesley (1954) (Translated from Russian)


Comments

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

How to Cite This Entry:
Attraction, partial domain of. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Attraction,_partial_domain_of&oldid=42266
This article was adapted from an original article by B.A. Rogozin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article