# Sticky central limit theorems on open books

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2013-12-01

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## Abstract

Given a probability distribution on an open book (a metric space obtained by gluing a disjoint union of copies of a half-space along their boundary hyperplanes), we define a precise concept of when the Fréchet mean (barycenter) is sticky. This nonclassical phenomenon is quantified by a law of large numbers (LLN) stating that the empirical mean eventually almost surely lies on the (codimension 1 and hence measure 0) spine that is the glued hyperplane, and a central limit theorem (CLT) stating that the limiting distribution is Gaussian and supported on the spine.We also state versions of the LLN and CLT for the cases where the mean is nonsticky (i.e., not lying on the spine) and partly sticky (i.e., is, on the spine but not sticky). © Institute of Mathematical Statistics, 2013.

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Hotz, Thomas, Stephan Huckemann, Huiling Le, JS Marron, Jonathan C Mattingly, Ezra Miller, James Nolen, Megan Owen, et al. (2013). Sticky central limit theorems on open books. *Annals of Applied Probability*, 23(6). pp. 2238–2258. 10.1214/12-AAP899 Retrieved from https://hdl.handle.net/10161/9519.

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### Scholars@Duke

#### Jonathan Christopher Mattingly

Jonathan Christopher Mattingly grew up in Charlotte, NC where he attended Irwin Ave elementary and Charlotte Country Day. He graduated from the NC School of Science and Mathematics and received a BS is Applied Mathematics with a concentration in physics from Yale University. After two years abroad with a year spent at ENS Lyon studying nonlinear and statistical physics on a Rotary Fellowship, he returned to the US to attend Princeton University where he obtained a PhD in Applied and Computational Mathematics in 1998. After 4 years as a Szego assistant professor at Stanford University and a year as a member of the IAS in Princeton, he moved to Duke in 2003. He is currently a Professor of Mathematics and of Statistical Science.

His expertise is in the longtime behavior of stochastic system including randomly forced fluid dynamics, turbulence, stochastic algorithms used in molecular dynamics and Bayesian sampling, and stochasticity in biochemical networks.

Since 2013 he has also been working to understand and quantify gerrymandering and its interaction of a region's geopolitical landscape. This has lead him to testify in a number of court cases including in North Carolina, which led to the NC congressional and both NC legislative maps being deemed unconstitutional and replaced for the 2020 elections.

He is the recipient of a Sloan Fellowship and a PECASE CAREER award. He is also a fellow of the IMS and the AMS. He was awarded the Defender of Freedom award by Common Cause for his work on Quantifying Gerrymandering.

#### Ezra Miller

Professor Miller's research centers around problems in geometry,

algebra, topology, probability, statistics, and computation

originating in mathematics and the sciences, including biology,

chemistry, computer science, and imaging.

The techniques range, for example, from abstract algebraic geometry or

commutative algebra of ideals and varieties to concrete metric or

discrete geometry of polyhedral spaces; from deep topological

constructions such as equivariant K-theory and stratified Morse theory

to elementary simplicial and persistent homology; from functorial

perspectives on homological algebra in the derived category to

specific constructions of complexes based on combinatorics of cell

decompositions; from geodesic collapse applied to central limit

theorems for samples from stratified spaces to dynamics of explicit

polynomial vector fields on polyhedra.

Beyond motivations from within mathematics, the sources of these

problems lie in, for example, graphs and trees in evolutionary biology

and medical imaging; mass-action kinetics of chemical reactions;

computational geometry, symbolic computation, and combinatorial game

theory; Lie theory; and geometric statistics of data sampled from

highly non-Euclidean spaces. Examples of datasets under consideration

include MRI images of blood vessels in human brains and lungs, 3D

folded protein structures, and photographs of fruit fly wings for

developmental morphological studies.

#### James H. Nolen

My research is in the area of probability and partial differential equations, which have been used to model many phenomena in the natural sciences and engineering. Asymptotic analysis has been a common theme in much of my research. Current research interests include: stochastic dynamics, interacting particle systems, reaction-diffusion equations, applications to biological models.

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