A note on asymptotics of linear combinations of IID random variables
Let \(X_{1},X_{2},\dots\) be i.i.d. random variables with the common distribution function \(F\). For \(n=1,2,\dots\), consider the linear combination \(S_{\boldsymbol{a}_{n}}=a_{n,1}X_{1}+a_{n,2}X_{2}+\cdots +a_{n,n}X_{n}\), where \(\boldsymbol{a}_{n}=\left(a_{n,1} ,a_{n,2}, \dots,a_{n,n}\right)\) is an arbitrary sequence of weights. The author investigates the asymptotic distribution of \(S_{\boldsymbol{a}_{n}}\) under the negligibility condition. He proves that, if \(S_{\boldsymbol{a}_{n}}\) is asymptotically normal, then the distribution \(F\) belongs to the domain of attraction of the 2-stable law.
scientific article; zbMATH DE number 5881206