On the joint distribution of the area and the number of peaks for Bernoulli excursions

Let $P_n$ be a random Bernoulli excursion of length $2n$. We show that the area under $P_n$ and the number of peaks of $P_n$ are asymptotically independent. We also show that these statistics have the correlation coefficient asymptotic to $c\mathrm{/}\sqrt{n}$ for large $n$, where $c<0$, and explicitly compute the coefficient $c$.