# Attraction, partial domain of

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)