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

TL;DR

This paper proves that for large Bernoulli excursions, the area and number of peaks are asymptotically independent, with their correlation diminishing as 1/√n, and explicitly calculates the correlation coefficient.

Contribution

It establishes the asymptotic independence of area and peaks in Bernoulli excursions and derives the explicit form of their correlation coefficient.

Findings

Area and peak count are asymptotically independent for large n.

Correlation coefficient decreases as c/√n, with c explicitly computed.

Provides a detailed asymptotic analysis of Bernoulli excursions.

Abstract

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 $c /\sqrt{n}$ for large $n$, where $c < 0$, and explicitly compute the coefficient $c$.