  We introduce a nonparametric way to estimate the global probability density
function for a random persistence diagram. Precisely, a kernel density function
centered at a given persistence diagram and a given bandwidth is constructed.
Our approach encapsulates the number of topological features and considers the
appearance or disappearance of features near the diagonal in a stable fashion.
In particular, the structure of our kernel individually tracks long persistence
features, while considering features near the diagonal as a collective unit.
The choice to describe short persistence features as a group reduces
computation time while simultaneously retaining accuracy. Indeed, we prove that
the associated kernel density estimate converges to the true distribution as
the number of persistence diagrams increases and the bandwidth shrinks
accordingly. We also establish the convergence of the mean absolute deviation
estimate, defined according to the bottleneck metric. Lastly, examples of
kernel density estimation are presented for typical underlying datasets.
