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We present new results for consistency of maximum likelihood estimators with a focus on multivariate mixed models. Our theory builds on the idea of using subsets of the full data to establish consistency of estimators based on the full…

We provide a new and simple system of equations for the normal sub-Riemannian geodesics. These use a partial connection that we show is canonically available, given a choice of complement to the distribution. We also describe conditions…

The classical random walk isomorphism theorems relate the local times of a continuous-time random walk to the square of a Gaussian free field. A Gaussian free field is a spin system that takes values in Euclidean space, and this article…

Statistical inference on the mean of a Poisson distribution is a fundamentally important problem with modern applications in, e.g., particle physics. The discreteness of the Poisson distribution makes this problem surprisingly challenging,…

Deep ensembles can be considered as the current state-of-the-art for uncertainty quantification in deep learning. While the approach was originally proposed as a non-Bayesian technique, arguments supporting its Bayesian footing have been…

We prove sharp estimates in a shrinking target problem for the action of an arbitrary subgroup $\Gamma$ of $SL_2(\mathbb{Z})$ on the 2-torus. This can also be viewed as a non-commutative Diophantine approximation problem. The methods…

The real Ginibre spherical ensemble consists of random matrices of the form $A B^{-1}$, where $A,B$ are independent standard real Gaussian $N \times N$ matrices. The expected number of real eigenvalues is known to be of order $\sqrt{N}$. We…

If a higher derivative theory arises from a transformation of variables that involves time derivatives, a tailor-made Hamiltonian formulation is shown to exist. The details and advantages of this elegant Hamiltonian formulation, which…

A spin-statistics theorem and a PCT theorem are obtained in the context of the superselection sectors in Quantum Field Theory on a 4-dimensional space-time. Our main assumption is the requirement that the modular groups of the von Neumann…

Spherical regression explores relationships between variables on spherical domains. We develop a nonparametric model that uses a diffeomorphic map from a sphere to itself. The restriction of this mapping to diffeomorphisms is natural in…

We develop a systematic framework for constructing spherical harmonics on the two-dimensional unit sphere as superpositions of Gaussian beams whose poles form well-separated point configurations. The distributional and analytic properties…

We strengthen the maximal ergodic theorem for actions of groups of polynomial growth to a form involving jump quantity, which is the sharpest result among the family of variational or maximal ergodic theorems. As a consequence, we deduce in…

An approximate statistical description of the formation and evolution of structure of the universe based on the Zel'dovich theory of gravitational instability is proposed. It is found that the evolution of DM structure shows features of…

A noetherian form is an abstract self-dual framework suitable for establishing homomorphism theorems (such as the isomorphism theorems and homological diagram lemmas) for group-like structures. In this paper we identify and carry out an…

We present a generic dynamic programming method to compute the optimal clustering of $n$ scalar elements into $k$ pairwise disjoint intervals. This case includes 1D Euclidean $k$-means, $k$-medoids, $k$-medians, $k$-centers, etc. We extend…

Statistical equilibrium models of coherent structures in two-dimensional and barotropic quasi-geostrophic turbulence are formulated using canonical and microcanonical ensembles, and the equivalence or nonequivalence of ensembles is…

Variance estimation is important for statistical inference. It becomes non-trivial when observations are masked by serial dependence structures and time-varying mean structures. Existing methods either ignore or sub-optimally handle these…

We propose a new look at the Julg-Valette theorem on K-theoretic amenability for groups operating on trees. The main tool is a generalization of a construction of uniformly bounded representations of the free groups due to Pytlik and…

We present a systematic account of supergravity theories in which the global scaling symmetry is gauged. This generalizes the standard gaugings of non-abelian off-shell symmetries. A particular feature of these theories is an additional…

Subsampling is commonly used to overcome computational and economical bottlenecks in the analysis of finite populations and massive datasets. Existing methods are often limited in scope and use optimality criteria (e.g., A-optimality) with…