随机立方过滤持久图的大偏差原理
Large deviation principle for persistence diagrams of random cubical filtrations
摘要：
本文的目的是研究当窗口大小趋于无穷大时，随机三次滤过的持久性图的渐近行为。这里，随机立方滤子是随机立方集的递增族，其是随机生成的具有欧几里得空间中的整数坐标的高维单位立方的并集。首先，我们证明了持久性图的强大数律，灵感来自平冈，白井，和Trinh的工作，在那里的持久性图的过滤随机几何复合物被认为是。相对于以前的文件处理持久性图的极限定理，本文旨在进一步研究持久性图的大偏差行为。本文证明了一类随机三次过滤的持久图的大偏差原理，并证明了其速率函数是极限对数矩母函数的Fenchel-Legendre变换.在证明中，我们还建立了一种将持久贝蒂数元组的大偏差原理提升到持久图的一般方法，以便广泛应用。
英文摘要：
The objective of this article is to investigate the asymptotic behavior of the persistence diagrams of a random cubical filtration as the window size tends to infinity. Here, a random cubical filtration is an increasing family of random cubical sets, which are the union of randomly generated higher-dimensional unit cubes with integer coordinates in a Euclidean space. We first prove the strong law of large numbers for the persistence diagrams, inspired by the work of Hiraoka, Shirai, and Trinh, where the persistence diagram of a filtration of random geometric complexes is considered. As opposed to prior papers treating limit theorems for persistence diagrams, the present article aims to further study the large deviation behavior of persistence diagrams. We prove a large deviation principle for the persistence diagrams of a class of random cubical filtrations, and show that the rate function is given as the Fenchel--Legendre transform of the limiting logarithmic moment generating function. In the proof, we also establish a general method of lifting a large deviation principle for the tuples of persistent Betti numbers to persistence diagrams for broad application.
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