Related papers: Generalized Limit Theorems For $U$-max Statistics
We study extremal statistics and return intervals in stationary long-range correlated sequences for which the underlying probability density function is bounded and uniform. The extremal statistics we consider e.g., maximum relative to…
Random developments of a path into a matrix Lie group $G_N$ have recently been used to construct signature-based kernels on path space. Two examples include developments into GL$(N;\mathbb{R})$ and $U(N;\mathbb{C})$, the general linear and…
Suppose X is a random vector, that is distributed uniformly in some n-dimensional convex set. It was conjectured that when the dimension n is very large, there exists a non-zero vector u, such that the distribution of the real random…
We completely characterize the class of univariate distributions allowing for a Stein kernel and illustrate our result by means of some concrete distributions. Moreover, we apply our findings to prove a quantitative version of the central…
We develop a simple and unified approach to investigate several aspects of the cluster statistics of random expansive (multi-)sets. In particular, we determine the limiting distribution of the size of the smallest and largest clusters, we…
We study the asymptotic behavior of the maximum interpoint distance of random points in a $d$-dimensional set with a unique diameter and a smooth boundary at the poles. Instead of investigating only a fixed number of $n$ points as $n$ tends…
In statistics on manifolds, the notion of the mean of a probability distribution becomes more involved than in a linear space. Several location statistics have been proposed, which reduce to the ordinary mean in Euclidean space. A…
In the setting where we have $n$ independent observations of a random variable $X$, we derive explicit error bounds in total variation distance when approximating the number of observations equal to the maximum of the sample (in the case…
Let $p, q \in (0, \infty]$ and $\ell_p^m(\ell_q^n)$ be the mixed-norm sequence space of real matrices $x = (x_{i, j})_{i \leq m, j \leq n}$ endowed with the (quasi-)norm $\Vert x \Vert_{p, q} := \big\Vert \big( \Vert (x_{i, j})_{j \leq n}…
Various methods in statistical learning build on kernels considered in reproducing kernel Hilbert spaces. In applications, the kernel is often selected based on characteristics of the problem and the data. This kernel is then employed to…
In this paper, we survey some recent progress on rigorously establishing the universality of various spectral statistics of Wigner Hermitian random matrix ensembles, focusing on the Four Moment Theorem and its refinements and applications,…
The notion of maximal-spacing in several dimensions was introduced and studied by Deheuvels (1983) for data uniformly distributed on the unit cube. Later on, Janson (1987) extended the results to data uniformly distributed on any bounded…
In this paper, we establish an exponential inequality for U-statistics of i.i.d. data, varying kernel and taking values in a separable Hilbert space. The bound are expressed as a sum of an exponential term plus an other one involving the…
We study the problem of distributional approximations to high-dimensional non-degenerate $U$-statistics with random kernels of diverging orders. Infinite-order $U$-statistics (IOUS) are a useful tool for constructing simultaneous prediction…
The distribution function of particles over clusters is proposed for a system of identical intersecting spheres, the centres of which are uniformly distributed in space. Consideration is based on the concept of the rank number of clusters,…
The law of large numbers for the empirical density for the pairs of uniformly distributed integers with a given greatest common divisor is a classic result in number theory. In this paper, we study the large deviations of the empirical…
In this paper, quantitative bounds in high-frequency central limit theorems are derived for Poisson based $U$-statistics of arbitrary degree built by means of wavelet coefficients over compact Riemannian manifolds. The wavelets considered…
We provide a general result for bounding the difference between point probabilities of integer supported distributions and the translated Poisson distribution, a convenient alternative to the discretized normal. We illustrate our theorem in…
We describe an approach to universality limits for orthogonal polynomials on the real line which is completely local and uses only the boundary behavior of the Weyl m-function at the point. We show that bulk universality of the…
The central limit theorem provides the theoretical foundation for the universality of the normal distribution: under broad conditions, the asymptotic distribution of a sum of independent random variables approaches a Gaussian. Yet, physical…