  We define a new topological summary for data that we call the persistence
landscape. Since this summary lies in a vector space, it is easy to combine
with tools from statistics and machine learning, in contrast to the standard
topological summaries. Viewed as a random variable with values in a Banach
space, this summary obeys a strong law of large numbers and a central limit
theorem. We show how a number of standard statistical tests can be used for
statistical inference using this summary. We also prove that this summary is
stable and that it can be used to provide lower bounds for the bottleneck and
Wasserstein distances.
