  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.
