Upper bounds for the probabilities $\mathbb{ P}(F\geq \mathbb{ E} F + r)$ and $\mathbb{ P}(F\leq \mathbb{ E} F - r)$ are proved, where $F$ is a certain component count associated with a random geometric graph built over a Poisson point process on $\mathbb{ R}^d$. The bounds for the upper tail decay exponentially, and the lower tail estimates even have a Gaussian decay. For the proof of the concentration inequalities, recently developed methods based on logarithmic Sobolev inequalities are used and enhanced. A particular advantage of this approach is that the resulting inequalities even apply in settings where the underlying Poisson process has infinite intensity measure.
Categories: math.PR