The remaining area of the convex hull of a Poisson process

TL;DR

This paper corrects the application of the time inversion argument in previous work on the remaining area of the convex hull of a Poisson process, providing accurate formulas for its expectation and variance.

Contribution

It clarifies and corrects the scaling constant in the variance formula by properly applying the time inversion argument.

Findings

Corrected the variance scaling constant for the remaining area.

Confirmed consistency with previous results by Nagaev and Khamdamov, and Buchta.

Provided accurate formulas for expectation and variance of the remaining area.

Abstract

In Cabo and Groeneboom (1994) the remaining area of the left-lower convex hull of a Poisson point process with intensity one in the first quadrant of the plane was analyzed, using the methods of Groeneboom (1988), giving formulas for the expectation and variance of the remaining area for a finite interval of slopes of the boundary of the convex hull. However, the time inversion argument of Groeneboom (1988) was not correctly applied in Cabo and Groeneboom (1994), leading to an incorrect scaling constant for the variance. The purpose of this note is to show how the correct application of the time inversion argument gives the right expression, which is in accordance with results in Nagaev and Khamdamov (1991) and Buchta (2003).