We study two-sample tests for relevant differences in persistence diagrams obtained from $L^p$-$m$-approximable data $(\mathcal{X}_t)_t$ and $(\mathcal{Y}_t)_t$. To this end, we compare variance estimates w.r.t.\ the Wasserstein metrics on the space of persistence diagrams. In detail, we consider two test procedures. The first compares the Fr{\'e}chet variances of the two samples based on estimators for the Fr{\'e}chet mean of the observed persistence diagrams $PD(\mathcal{X}_i)$ ($1\le i\le m$), resp., $PD(\mathcal{Y}_j)$ ($1\le j\le n$) of a given feature dimension. We use classical functional central limit theorems to establish consistency of the testing procedure. The second procedure relies on a comparison of the so-called independent copy variances of the respective samples. Technically, this leads to functional central limit theorems for U-statistics built on $L^p$-$m$-approximable sample data.

翻译：我们研究了从$L^p$-$m$可逼近数据$(\mathcal{X}_t)_t$和$(\mathcal{Y}_t)_t$获得的持久化图中相关差异的双样本检验。为此，我们比较了持久化图空间上关于Wasserstein度量的方差估计。具体而言，我们考虑了两种检验程序。第一种是基于给定特征维度的观测持久化图$PD(\mathcal{X}_i)$ ($1\le i\le m$)和$PD(\mathcal{Y}_j)$ ($1\le j\le n$)的Fr\'echet均值估计，比较两个样本的Fr\'echet方差。我们利用经典泛函中心极限定理来建立该检验程序的一致性。第二种程序依赖于比较各自样本的所谓独立副本方差。从技术上讲，这导致了对构建于$L^p$-$m$可逼近样本数据上的U统计量的泛函中心极限定理。