On Weak Convergence of Moments of the Sum of Independently and Identically Distributed Random Variables

We study the behavior of the “moments” $\chi_n(\varepsilon) = \mathbf{E} g(|S_n|)\,\frac{\mathbf{I}[|S_n|\ge\varepsilon a_n]}{g(a_n)}$, $\varepsilon>0$, where $S_n$ is the sum of $n$ independent copies of some random variable, the sequence $\{ a_n, \, n\ge 1\}$ is monotone increasing to infinity, and the positive function $g(y)$ satisfies $g(y)\nearrow$, $g(u)/u^\gamma\searrow\,$, $y>y_0$ (with some $\gamma>0)$. In particular, the following result is proved: $\chi_n(\varepsilon) = o(1)$ (for each $\varepsilon>0$) if and only if $n\,\mathbf{E} X\,\frac{\mathbf I[|X|<a_n]}{a_n}\to 0$, $n\,\mathbf{E} X^2\,\frac{\mathbf I[|X|<a_n]}{a^2_n} \to 0$, and $n\, \mathbf{E} g(|X|)\,\frac{\mathbf I[|X|\ge a_n]}{g(a_n)} = o(1)$.