This open access book presents the key aspects of statistics in Wasserstein spaces i.e.
statistics in the space of probability measures when endowed with the geometry of optimal
transportation. Further to reviewing state-of-the-art aspects it also provides an accessible
introduction to the fundamentals of this current topic as well as an overview that will serve
as an invitation and catalyst for further research. Statistics in Wasserstein spaces represents
an emerging topic in mathematical statistics situated at the interface between functional data
analysis (where the data are functions thus lying in infinite dimensional Hilbert space) and
non-Euclidean statistics (where the data satisfy nonlinear constraints thus lying on
non-Euclidean manifolds). The Wasserstein space provides the natural mathematical formalism to
describe data collections that are best modeled as random measures on Euclidean space (e.g.
images and point processes). Such random measures carry the infinite dimensional traits of
functional data but are intrinsically nonlinear due to positivity and integrability
restrictions. Indeed their dominating statistical variation arises through random deformations
of an underlying template a theme that is pursued in depth in this monograph.
