Weak Convergence of a Certain Functional

We consider the functional $T_n=(S_1^2+\dots+S_n^2)/(nV_n^2)$ derived from a sequence $\{X_n\}_{n\ge 1}$ of independent identically distributed random variables, where $S_k=X_1+\dots+X_k$, $V_n^2=X_1^2+\dots+X_n^2$. Let $G$ be the distribution function of the random variable $\int_{0}^{1}W^2(t)\,dt$, where $W(t)$, $t\in [0,1]$, is a Wiener process. We show that the distribution function $T_n$ weakly converges to $G$ as $n\to\infty$ if and only if the distribution function of the random variable $X_1$ belongs to the attraction domain of the normal law and $\mathbf{E}X_1=0$.