부산대학교 도서관

Let $\xi$ be a random variable with the density $f$, mean $\mu$ and variance $\sigma^2$. Denote by $w(x)\equiv w_\xi(x)$ the solution of the equation $\sigma^2 w(x)f(x)=\int^x_{-\infty}(\mu-t)f(t)\,dt$, and let $Z_\xi=w(\xi)$. \par It is shown that $EZ^2_\xi\geq 1$, with equality if and only if $\xi$ is normally distributed. Moreover, it is proved that if $\{\zeta_n\}$ is a sequence of random variables such that $EZ^2_{\zeta n}\to 1$ as $n\to\infty$, then $(\zeta_n-E\zeta_n)(D\zeta_n)^{-1/2}$ is asymptotically standard normal. On the basis of these assertions the authors obtain the proof of the central limit theorem for a sequence of independent identically distributed random variables.