If $X_1, X_2,\ldots $ are i.i.d Bernoulli random variables with mean $\frac{1}{2}$. Define $T_n = \sqrt{n}(\frac{4\sum_{i=1}^{n} X_i - 2n}{\sum_{i=1}^{n} X_i^2})$. Find the pmf or pdf of the limiting distribution of the sequence $T_1, T_2,\ldots$
My thought: We consider the function $g(x) = 4x$. Since this function and its derivative are continuous, and both are nonzero at $u_x = \frac{1}{2}$, we could apply the theorem saying $\sqrt{n}[\frac{g(\overline{X_n})- g(\frac{1}{2})}{|g'(\frac{1}{2})\sigma_{x}}]\rightarrow N(0,1)$ in distribution, where $X_n = \frac{X_1+\ldots + X_n}{n}$ and $\sigma_{x} = \frac{1}{4}$ (since $X$ is Bernoulli). Now, my plan is to figure out the convergence in probability of $\sum_{i=1}^{n} X_i^2$, but I am stucked here.
Could anyone please help with this last piece of puzzle?
You can use the law of large numbers to talk about what happens to $\frac{1}{n} \sum_{i=1}^n X_i^2$. Note that you divided the numerator of $T_n$ by $n$, so you should also do that for the denominator.