  We develop a general framework for the probabilistic analysis of random
finite point clouds in the context of topological data analysis. We extend the
notion of a barcode of a finite point cloud to compact metric spaces. Such a
barcode lives in the completion of the space of barcodes with respect to the
bottleneck distance, which is quite natural from an analytic point of view. As
an application we prove that the barcodes of i.i.d. random variables sampled
from a compact metric space converge to the barcode of the support of their
distribution when the number of points goes to infinity. We also examine more
quantitative convergence questions for uniform sampling from compact manifolds,
including expectations of transforms of barcode valued random variables in
Banach spaces. We believe that the methods developed here will serve as useful
tools in studying more sophisticated questions in topological data analysis and
related fields.
