Let $X_1,X_2...X_n \sim U(0,1)$ be $i.i.d $, $S_n=\sum_{m=1}^{n}X_m$, please find the limiting distribution of $\sum_{m=1}^{n} \mathbb I_{X_m S_n\leq1}$.
I guess that might be Poisson distribution. And Let $Y_{m,n}= \mathbb I_{X_m S_n\leq1}$ and want to verify the two conditions for Possion convergence theorem. The first condition is that we need $\sum_{m=1}^{n}p_{m,n} \to \lambda.$ Actually $p_{m,n} \leq P(X_m\leq \frac{1}{(\frac{1}{2}-\delta)n})+P(\frac{S_n}{n}\leq \frac{1}{2}-\delta)$, when summation, I have no idea how to control the second term. Can you give me some hint or another method?