I am stuck at the following exercise:

Let $ (X_k)_k$ be a sequence of i.i.d. Bernoulli$ (p)$ distributed RVs with $ p=1/2$ . We want to estimate $ Var(X_1) = p(1-p) = p^2 =: \tau_p^2$ by directly plugging in the natural estimator $ \widetilde{p} := \frac{1}{n}\sum_{i=1}^n X_i$ of $ p$ . We call this estimator for the variance $ \widetilde{\tau}^2_{p,n}$ . Examine the limit distribution of $ \widetilde{\tau}^2_{p,n}$ .

I do not see how I could do this. I recognise that we are basically looking at a uniform distribution of two events and I see that therefore we have

$ $ \widetilde{\tau}^2_{p,n} = \frac{1}{n^2} \biggl(\sum_{i=1}^n X_i \biggr) ^2$ $

, but I do not see how to proceed now. Intuitively I would say that $ \widetilde{\tau}^2_{p,n}$ should also be uniformly distributed, but how can I prove this?