Related papers: Approximation of Beta-Jacobi ensembles by Beta-Lag…

We establish some exact asymptotic results for a matching problem with respect to a family of beta distributions. Let $X_1, \ldots, X_n$ be independent random variables with common distribution the symmetric Jacobi measure $d\mu (x) = C_d…

We show that the Subgroup distance problem regarding the Hamming distance, the Cayley distance and the $l_\infty$ distance is NP-complete when the input group is cyclic. When we restrict the $l_\infty$ distance to fixed values we show that…

For the random eigenvalues with density corresponding to the Jacobi ensemble $$c \cdot \prod_{i < j} | \lambda_i - \lambda_j |^\beta \prod^n_{i=1} (2 - \lambda_i)^a (2 + \lambda_i)^b I_{(-2,2)} (\lambda_i) $$ $(a, b > -1, \beta > 0) $ a…

The $\alpha$-divergences include the well-known Kullback-Leibler divergence, Hellinger distance and $\chi^2$-divergence. In this paper, we derive differential and integral relations between the $\alpha$-divergences that are generalizations…

Bernstein-von Mises results (BvM) establish that the Laplace approximation is asymptotically correct in the large-data limit. However, these results are inappropriate for computational purposes since they only hold over most, and not all,…

We determine the distance (up to a multiplicative constant) in the Zygmund class $\Lambda_{\ast}(\mathbb{R}^n)$ to the subspace $\mathrm{J}(\mathbf{bmo})(\mathbb{R}^n).$ The latter space is the image under the Bessel potential $J :=…

Let $\theta_1,\ldots,\theta_n$ be random variables from Dyson's circular $\beta$-ensemble with probability density function $\operatorname {Const}\cdot\prod_{1\leq j<k\leq n}|e^{i\theta_j}-e^{i\theta _k}|^{\beta}$. For each $n\geq2$ and…

We consider vectors from $\{0,1\}^n$. The weight of such a vector $v$ is the sum of the coordinates of $v$. The distance ratio of a set $L$ of vectors is ${\rm dr}(L):=\max \{\rho(x,y):\ x,y \in L\}/ \min \{\rho(x,y):\ x,y \in L,\ x\neq…

We consider a ${\Lambda}$-coalescent and we study the asymptotic behavior of the total length $L^{(n)}_{ext}$ of the external branches of the associated $n$-coalescent. For Kingman coalescent, i.e. ${\Lambda}={\delta}_0$, the result is well…

We study the Kullback--Leibler (KL) divergence approximation theory of Gaussian mixture models (GMMs) by isolating an abstract mechanism behind several necessary-and-sufficient statements. The necessity direction is universal: if a density…

Recently, a new distance has been introduced for the graphs of two point-to-set operators, one of which is maximally monotone. When both operators are the subdifferential of a proper lower semicontinuous convex function, this distance…

In this paper, we discuss a property of the Kullback--Leibler divergence measured between two models of the family of the location-scale distributions. We show that, if model $M_1$ and model $M_2$ are represented by location-scale…

Let the space $\mathbb{R}^n$ be endowed with a Minkowski structure $M$ (that is $M\colon \mathbb{R}^n \to [0,+\infty)$ is the gauge function of a compact convex set having the origin as an interior point, and with boundary of class $C^2$),…

The purpose of this paper is twofold. On a technical side, we propose an extension of the Hausdorff distance from metric spaces to spaces equipped with asymmetric distance measures. Specifically, we focus on the family of Bregman…

We establish that the set of pairs $(\alpha, \beta)$ of real numbers such that $$ \liminf_{q \to + \infty} q \cdot (\log q)^2 \cdot \Vert q \alpha \Vert \cdot \Vert q \beta \Vert > 0, $$ where $\Vert \cdot \Vert$ denotes the distance to the…

The $\beta$-ensembles of random matrix theory with classical weights have many special properties. One is that the loop equations specifying the resolvent and corresponding multipoint correlators permit a derivation at general order of the…

This paper shows that large nonparametric classes of conditional multivariate densities can be approximated in the Kullback--Leibler distance by different specifications of finite mixtures of normal regressions in which normal means and…

We consider the problem of finding commuting self-adjoint extensions of the partial derivatives {(1/i)(\partial/\partial x_j):j=1,...,d} with domain C_c^\infty(\Omega) where the self-adjointness is defined relative to L^2(\Omega), and…

Let $\varepsilon>0$. We construct an explicit, full-measure set of $\alpha \in[0,1]$ such that if $\gamma \in \mathbb{R}$ then, for almost all $\beta \in[0,1]$, if $\delta \in \mathbb{R}$ then there are infinitely many integers $n\geq 1$…

For $i = 0, 1, 2, \dots, k$, let $\mu_i$ be a Borel probability measure on $[0,1]$ which is equivalent to Lebesgue measure $\lambda$ and let $T_i:[0,1] \rightarrow [0,1]$ be $\mu_i$-preserving ergodic transformations. We say that…