Asymptotic behavior of lifetime sums for random simplicial complex processes

Minimum spanning acycle and lifetime of persistent homology in the Linial-Meshulam process

This paper studies a higher dimensional generalization of Frieze's $ζ(3)$-limit theorem in the Erdös-Rényi graph process. Frieze's theorem states that the expected weight of the minimum spanning tree converges to $ζ(3)$ as the number of vertices goes to infinity. In this paper, we study the $d$-Linial-Meshulam process as a model for random simplicial complexes, where $d=1$ corresponds to the Erdös-Rényi graph process. First, we define spanning acycles as a higher dimensional analogue of spanning trees, and connect its minimum weight to persistent homology. Then, our main result shows that the …

多参数复过程持续同调的极限定理

Limit Theorems for Persistent Homology of Multiparameter Complex Processes

本文针对基于Costa-Farber模型的多参数复过程，建立其持续Betti数的大数定律，证明持续图依概率弱收敛到确定性有限测度，并推导了相关测度在最优传输距离下的收敛性。

Weighted Persistent Homology Sums of Random Čech Complexes

We study the asymptotic behavior of random variables of the form \begin{equation*} E_α^i\left(x_1,\ldots,x_n\right)=\sum_{\left(b,d\right)\in \mathit{PH}_i\left(x_1,\ldots,x_n\right)} \left(d-b\right)^α \end{equation*} where $\left\{x_j\right\}_{j\in\mathbb{N}}$ are i.i.d. samples from a probability measure on a triangulable metric space, and $\textit{PH}_i\left(x_1,\ldots,x_n\right)$ denotes the $i$-dimensional reduced persistent homology of the Čech complex of $\left\{x_1,\ldots,x_n\right\}.$ These quantities are a higher-dimensional generalization of the $α$-weighted sum of a minimal spanni…

The $β$-Delaunay tessellation II: The Gaussian limit tessellation

We study the weak convergence of $β$- and $β'$-Delaunay tessellations in $\mathbb{R}^{d-1}$ that were introduced in part I of this paper, as $β\to\infty$. The limiting stationary simplicial random tessellation, which is called the Gaussian-Delaunay tessellation, is characterized in terms of a space-time paraboloid hull process in $\mathbb{R}^{d-1}\times\mathbb{R}$. The latter object has previously appeared in the analysis of the number of shocks in the solution of the inviscid Burgers' equation and the description of the local asymptotic geometry of Gaussian random polytopes. In this paper it …

Limit theorems for random cubical homology

This paper studies random cubical sets in $\mathbb{R}^d$. Given a cubical set $X\subset \mathbb{R}^d$, a random variable $ω_Q\in[0,1]$ is assigned for each elementary cube $Q$ in $X$, and a random cubical set $X(t)$ is defined by the sublevel set of $X$ consisting of elementary cubes with $ω_Q\leq t$ for each $t\in[0,1]$. Under this setting, the main results of this paper show the limit theorems (law of large numbers and central limit theorem) for Betti numbers and lifetime sums of random cubical sets and filtrations. In addition to the limit theorems, the positivity of the limiting Betti numb…

Generalized negative binomial distributions as mixed geometric laws and related limit theorems

In this paper we study a wide and flexible family of discrete distributions, the so-called generalized negative binomial (GNB) distributions that are mixed Poisson distributions in which the mixing laws belong to the class of generalized gamma (GG) distributions. The latter was introduced by E. W. Stacy as a special family of lifetime distributions containing gamma, exponential power and Weibull distributions. These distributions seem to be very promising in the statistical description of many real phenomena being very convenient and almost universal models for the description of statistical r…