  The prevailing statistical approach to analyzing persistence diagrams is
concerned with filtering out topological noise. In this paper, we adopt a
different viewpoint and aim at estimating the actual distribution of a random
persistence diagram, which captures both topological signal and noise. To that
effect, Chazel and Divol (2019) proved that, under general conditions, the
expected value of a random persistence diagram is a measure admitting a
Lebesgue density, called the persistence intensity function. In this paper, we
are concerned with estimating the persistence intensity function and a novel,
normalized version of it -- called the persistence density function. We present
a class of kernel-based estimators based on an i.i.d. sample of persistence
diagrams and derive estimation rates in the supremum norm. As a direct
corollary, we obtain uniform consistency rates for estimating linear
representations of persistence diagrams, including Betti numbers and
persistence surfaces. Interestingly, the persistence density function delivers
stronger statistical guarantees.
