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We propose a high dimensional mean test framework for shrinking random variables, where the underlying random variables shrink to zero as the sample size increases. By pooling observations across overlapping subsets of dimensions, we…

This paper studies Kernel Density Estimation for a high-dimensional distribution $\rho(x)$. Traditional approaches have focused on the limit of large number of data points $n$ and fixed dimension $d$. We analyze instead the regime where…

We present a comprehensive analysis of an algorithm for evaluating high-dimensional polynomials that are invariant under permutations and rotations. The key bottleneck is the contraction of a high-dimensional symmetric and sparse tensor…

We describe a new MCMC method optimized for the sampling of probability measures on Hilbert space which have a density with respect to a Gaussian; such measures arise in the Bayesian approach to inverse problems, and in conditioned…

We study the problem of testing discrete distributions with a focus on the high probability regime. Specifically, given samples from one or more discrete distributions, a property $\mathcal{P}$, and parameters $0< \epsilon, \delta <1$, we…

We prove that in a metric measure space $X$, if for some $p \in (1,\infty)$ there are uniform bounds (independent of the measure) for the weak type $(p,p)$ of the centered maximal operator, then $X$ satisfies a certain geometric condition,…

We develop a thermodynamic formalism for quasi-multiplicative potentials on a countable symbolic space and apply these results to the dimension theory of infinitely generated self-affine sets. The first application is a generalisation of…

Let $X$ be a complex manifold, and let $Y$ and $D$ be two reduced simple-normal-crossing (snc) divisors on $X$ with no common irreducible components. Given a proper locally K\"ahler morphism $\pi \colon X \to \Delta$ from $X$ to a complex…

We establish a new global endpoint Sobolev inequality for measures that extends the classical theorem of Meyers-Ziemer by placing a maximal function on the right-hand side. This result has several significant consequences. It extends…

Li and Hu recently established variance-type O(1/n) bounds for the sample mean of independent random vectors under sublinear expectations. We extend their results to the exponential concentration regime. For bounded, independent R^d-valued…

We study finitely additive measures on the set $\mathbb N$ which extend the asymptotic density (density measures). We show that there is a one-to-one correspondence between density measures and positive functionals in $\ell_\infty^*$, which…

For a given transcendental number $\xi$ and for any polynomial $P(X)=: \lambda_0+\cdots+\lambda_k X^k \in \mathbb{Z}[X]$, we know that $ P(\xi) \neq 0.$ Let $k \geq 1$ and $\omega (k, H)$ be the infimum of the numbers $r > 0$ satisfying the…

We prove that if $E\subseteq \R^2$ is analytic and $1<d < \dim_H(E)$, there are ``many'' points $x\in E$ such that the Hausdorff dimension of the pinned distance set $\Delta_x E$ is at least $d\left(1 -…

The knockoffs is a recently proposed powerful framework that effectively controls the false discovery rate (FDR) for variable selection. However, none of the existing knockoff solutions are directly suited to handle multivariate or…

The purpose of this note is to record a consequence, for general metric spaces, of a recent result of David Bate. We prove the following fact: Let $X$ be a compact metric space of topological dimension $n$. Suppose that the $n$-dimensional…

We study approaches for compressing the empirical measure in the context of finite dimensional reproducing kernel Hilbert spaces (RKHSs). In this context, the empirical measure is contained within a natural convex set and can be…

By classical Fatou type theorems in various setups, it is well-known that positive harmonic functions have non-tangential limit at almost every point on the boundary. In this paper, in the setting of non-positively curved Harmonic manifolds…

We characterize the model spaces $K_\Theta$ in which functions with smooth boundary extensions are dense. It is shown that such approximations are possible if and only if the singular measure associated to the singular inner factor of…

Let $X_1,\ldots,X_n$ be independent identically distributed random vectors in $\mathbb{R}^d$. We consider upper bounds on $\max_x \mathbb{P}(a_1X_1+\cdots+a_nX_n=x)$ under various restrictions on $X_i$ and the weights $a_i$. When…

We develop a general method to bound the spreading of an entire wavepacket under Schr\"odinger dynamics from above. This method derives upper bounds on time-averaged moments of the position operator from lower bounds on norms of transfer…