Title: Concentration inequalities and large deviations for continuous greedy animals and paths Show Chinese title

Title: 连续贪心动物和路径的集中不等式和大偏差

Abstract: Consider the continuous greedy paths model: given a $d$-dimensional Poisson point process with positive marks interpreted as masses, let $\mathrm P(\ell)$ denote the maximum mass gathered by a path of length $\ell$ starting from the origin. It is known that $\mathrm P(\ell)/\ell converges a.s.\ to a deterministic constant $\mathrm P$. We show that the lower-tail deviation probability for $\mathrm P(\ell) has order $\mathrm{exp}(-\ell^2)$ and, under exponential moment assumption on the mass distribution, that the upper-tail deviation probability has order $\mathrm{exp}(-\ell)$. In the latter regime, we prove the existence and some properties -notably, convexity -of the corresponding rate function. An immediate corollary is the large deviation principle at speed $\ell$ for $\mathrm P(\ell)$. Along the proof we show an upper-tail concentration inequality in the case where marks are bounded. All of the above also holds for greedy animals and have versions where the paths or animals involved have two anchors instead of one. Show Chinese abstract

Abstract: 考虑连续贪婪路径模型：给定一个$d$维的带有正标记的泊松点过程，这些标记被解释为质量，令$\mathrm P(\ell)$表示从原点出发长度为$\ell$的路径所收集的最大质量。 已知$\mathrm P(\ell)/\ell converges a.s.\ to a deterministic constant $\mathrm P$. We show that the lower-tail deviation probability for $\mathrm P(\ell ) 的阶为$\mathrm{exp}(-\ell^2)$，并且在质量分布满足指数矩假设的情况下，上尾偏离概率的阶为$\mathrm{exp}(-\ell)$。 在后一种情形下，我们证明了相应速率函数的存在性及其一些性质——特别是凸性。 一个直接的推论是，在速度$\ell$下，对于$\mathrm P(\ell)$的大偏差原理成立。 在证明过程中，我们展示了当标记有界时上尾集中不等式。 以上所有结果也适用于贪婪动物，并且存在版本中涉及的路径或动物有两个锚点而不是一个。