On the Norming Constants in the Feller-Khintchine-Lévy Central Limit Theorem

Abstract

In the Feller-Khintchine-Levy central limit theorem for sums of i,i.d. random variables $\{X_n,\hspace{0.17em}\hspace{0.17em}n\hspace{0.17em}\hspace{0.17em}⩾\hspace{0.17em}\hspace{0.17em}1\}\hspace{0.17em}\hspace{0.17em}\mathrm{with}\hspace{0.17em}\hspace{0.17em}E{X}_{\hspace{0.17em}1}^2=\infty$ and the function $G(x)={\int }_o^x\hspace{0.17em}2tP\hspace{0.17em}\hspace{0.17em}\left\{\hspace{0.17em}|\hspace{0.17em}{X}_1\hspace{0.17em}\hspace{0.17em}|\hspace{0.17em}\hspace{0.17em}>\hspace{0.17em}\hspace{0.17em}t\right\}$ dt slowly varying, conditions are given for the norming sequence {B_n, n ⩾ 1} to have the explicit asymptotic representation $B_n\sim \sqrt{nG\left(\sqrt{n}\right)}.$