Let $I_A=1$ if the condition $A$ holds and $0$ otherwise.
Let $ X_1, X_2, \ldots $ be iid Bernoulli($p$) random variables, that is, $P(X_i=1)=p$ and $P(X_i=0)=1-p$.
Let $\displaystyle T_n=\sum_{i=1}^nI_{(X_i=1,X_{i+1}=1)}$.
Show that $\sqrt{n} \left(\frac{T_n}{n}-p^2\right)\overset{D}{\rightarrow} N(0, \sigma^2)$ and determine $\sigma^2$.
I can show that $E(T_n)=p^2$ and $Var(T_n)=p^2+2p^3-3p^4$ but I have difficulty for proving limiting distribution.