Let $X_n \sim U[0,1]$. Let $A_n$ count the number of local maxima of the sequence unto $n$. Prove a suitable central limit theorem for $A_n$.

Question

Let $X_n $ be uniformly distributed on $[0,1]$. We say $X_k$ is a local maximum if $X_k> X_{k\pm 1}$. Let $A_n$ count the number of local maxima of the sequence unto and including $n$. Find $a_n, b_n$ such that

$$\frac{A_n-a_n}{b_n} \longrightarrow N(0,1)$$

If someone could give a hint on how to approach this problem that would be great. I understand that I will need to use the Lindberg central limit theorem at some point to show the convergence.