  In topological data analysis, persistent homology characterizes robust
topological features in data and it has a summary representation, called a
persistence diagram. Statistical research for persistence diagrams have been
actively developed, and the persistence weighted kernel shows several
advantages over other statistical methods for persistence diagrams. If data is
drawn from some probability distribution, the corresponding persistence diagram
have randomness. Then, the expectation of the persistence diagram by the
persistence weighted kernel is well-defined. In this paper, we study
relationships between a probability distribution and the persistence weighted
kernel in the viewpoint of (1) the strong law of large numbers and the central
limit theorem, (2) a confidence interval to estimate the expectation of the
persistence weighted kernel numerically, and (3) the stability theorem to
ensure the continuity of the map from a probability distribution to the
expectation. In numerical experiments, we demonstrate our method gives an
interesting counterexample to a common view in topological data analysis.
