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A note on asymptotics of linear combinations of IID random variables. (English) Zbl 1224.60033
Let ${X}_{1},{X}_{2},\cdots$ be i.i.d. random variables with the common distribution function $F$. For $n=1,2,\cdots$, consider the linear combination ${S}_{{a}_{n}}={a}_{n,1}{X}_{1}+{a}_{n,2}{X}_{2}+\cdots +{a}_{n,n}{X}_{n}$, where ${a}_{n}=\left({a}_{n,1},{a}_{n,2},\cdots ,{a}_{n,n}\right)$ is an arbitrary sequence of weights. The author investigates the asymptotic distribution of ${S}_{{a}_{n}}$ under the negligibility condition. He proves that, if ${S}_{{a}_{n}}$ is asymptotically normal, then the distribution $F$ belongs to the domain of attraction of the 2-stable law.
##### MSC:
 60F05 Central limit and other weak theorems
##### References:
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