  Persistence diagrams are useful displays that give a summary information
regarding the topological features of some phenomenon. Usually, only one
persistence diagram is available, and replicated persistence diagrams are
needed for statistical inference. One option for generating these replications
is to fit a distribution for the points on the persistence diagram. The type of
the relevant distribution depends on the way the persistence diagram is
builded. There are two approaches for building the persistence diagram, one is
based on the Vietoris-Rips complex, and the second is based on some fitted
function such as the kernel density estimator. The two approaches yield a two
dimensional persistence diagram, where the coordinates of each point are the
'birth' and 'death' times. For the first approach, however, the 'birth' time is
zero for all the points that present the connected components of the
phenomenon. In this paper we examine the distribution of the connected
components when the persistence diagram is based on Vietoris-Rips complex. In
addition, we study the behaviour of the connected components when the
phenomenon is measured with noise.
