The central limit theorem (CLT) is commonly thought of as
occurring on the real line, or in multivariate form on a
real vector space. Motivated by statistical applications
involving nonlinear data, such as angles or phylogenetic
trees, the past twenty years have seen CLTs proved for
Fréchet means on manifolds and on certain examples of
singular spaces built from flat pieces glued together in
combinatorial ways. These CLTs reduce to the linear case
by tangent space approximation or by gluing. What should a
CLT look like on general non-smooth spaces, where tangent
spaces are not linear and no combinatorial gluing or flat
pieces are available? Answering this question involves
figuring out appropriate classes of spaces and measures,
correct analogues of Gaussian random variables, how the
geometry of the space (think "curvature") is reflected in
the limiting distribution, and generally how the geometry
of sampling from measures on singular spaces behaves. This
talk provides an overview of these answers, starting with a
review of the usual linear CLT and its generalization to
smooth manifolds, viewed through a lens that casts the
singular CLT as a natural outgrowth, and concluding with
how this investigation opens gateways to further advances
in geometric probability, topology, and statistics. Joint
work with Jonathan Mattingly and Do Tran.
