Limiting variance of partial uniform sum

Let $X_1, X_2, \dots$ be a sequence of i.i.d. random variables each having $U(0, 1)$ distribution. Let $Y$ be a random variable having the distribution function $G$. Suppose that $$\lim_{n\to\infty} P\left(\frac{X_1+X_2+X_n}{4} \le x\right) = G(x),\quad\text{for all }x\in\mathbb{R}.$$ Then, $\mathrm{Var}(Y)$ equals