  The persistent homology of a stationary point process on ${\bf R}^N$ is
studied in this paper. As a generalization of continuum percolation theory, we
study higher dimensional topological features of the point process such as
loops, cavities, etc. in a multiscale way. The key ingredient is the
persistence diagram, which is an expression of the persistent homology. We
prove the strong law of large numbers for persistence diagrams as the window
size tends to infinity and give a sufficient condition for the limiting
persistence diagram to have the full support. We also discuss a central limit
theorem for persistent Betti numbers.
