Concentration for Poisson functionals: Component counts in random geometric graphs

详细信息    查看全文

• 作者：Sascha Bachmann ^{sascha.bachmann@uni-osnabrueck.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 60D05 ; secondary ; 05C80 ; 60C05
• 刊名：Stochastic Processes and their Applications
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：126
• 期：5
• 页码：1306-1330
• 全文大小：382 K

文摘

Upper bounds for the probabilities P(F≥EF+r)$\mathbb{P}(F\ge \mathbb{E}F+r)$ and P(F≤EF−r)$\mathbb{P}(F\le \mathbb{E}F-r)$ are proved, where F$F$ is a certain component count associated with a random geometric graph built over a Poisson point process on R^d${\mathbb{R}}^d$. 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.