Tight Lower Bounds on the Contact Distance Distribution in Poisson Hole Process

摘要

In this letter, we derive new lower bounds on the cumulative distributionfunction (CDF) of the contact distance in the Poisson Hole Process (PHP) fortwo cases: (i) reference point is selected uniformly at random from$\mathbb{R}^2$ independently of the PHP, and (ii) reference point is located atthe center of a hole selected uniformly at random from the PHP. While one canderive upper bounds on the CDF of contact distance by simply ignoring theeffect of holes, deriving lower bounds is known to be relatively morechallenging. As a part of our proof, we introduce a tractable way of boundingthe effect of all the holes in a PHP, which can be used to study otherproperties of a PHP as well.