Related papers: Generalized Bounded Variation and Inserting point …

We study the quantization errors for the doubling probability measures $\mu$ which are supported on a class of Moran sets $E\subset\mathbb{R}^q$. For each $n\geq 1$, let $\alpha_n$ be an arbitrary $n$-optimal set for $\mu$ of order $r$ and…

Let $G$ be a countable discrete amenable group which acts continuously on a compact metric space $X$ and let $\mu$ be an ergodic $G-$invariant Borel probability measure on $X$. For a fixed tempered F{\o}lner sequence $\{F_n\}$ in $G$ with…

Let $n$ be a large integer, and let $G$ be the standard Gaussian vector in $R^n$. Paouris, Valettas and Zinn (2015) showed that for all $p\in[1,c\log n]$, the variance of the $\ell_p^n$--norm of $G$ is equivalent, up to a constant multiple,…

The classical Dvoretzky covering problem asks for conditions on the sequence of lengths $\{\ell_n\}_{n\in \mathbb{N}}$ so that the random intervals $I_n : = (\omega_n -(\ell_n/2), \omega_n +(\ell_n/2))$ where $\omega_n$ is a sequence of…

This paper provides a complete proof of Simon-Lukic conjecture for orthogonal polynomials on the unit circle. For a probability measure $d\mu = w(\theta) \frac{d\theta}{2\pi} + d\mu_s$ with Verblunsky coefficients…

For a sample of absolutely bounded i.i.d. random variables with a continuous density the cumulative distribution function of the sample variance is represented by a univariate integral over a Fourier series. If the density is a polynomial…

Given a probability measure $\mu$ on the unit circle $\mathbb{T}$, consider the reproducing kernel $k_{\mu,n}(z_1, z_2)$ in the space of polynomials of degree at most $n-1$ with the $L^2(\mu)$-inner product. Let $u, v \in \mathbb{C}$. It is…

Let $G$ denotes a finite abelian group of order $n$ and Davenport constant $D$, and put $m= n+D-1$. Let $x=(x_1, ..., x_m)\in G^m$ be a sequence with a maximal repetition $\ell$ attained by $x_m$ and put $r=\min(D,\ell)$. Let $w=(w_1, ...,…

Let $\beta >1$ be a non-integer. We consider expansions of the form $\sum_{i=1}^{\infty} d_i \beta^{-i}$, where the digits $(d_i)_{i \geq 1}$ are generated by means of a Borel map $K_{\beta}$ defined on $\{0,1\}^{\N}\times [ 0, \lfloor…

Let $\mu$ be a probability measure on $\mathbb{Z}$ that is not a Dirac mass and that has finite support. We prove that if the coefficients of a monic polynomial $f(x)\in\mathbb{Z}[x]$ of degree $n$ are chosen independently at random…

Let $(M,g)$ be a compact, connected Riemannian manifold of dimension $n\ge 2$, and let $\{e_j\}_{j=0}^\infty$ be an orthonormal basis of Laplace eigenfunctions $-\Delta_g e_j=\lambda_j^2 e_j$. Given a finite Borel measure $\mu$ on $M$,…

We show that probability measures on the unit circle associated with Verblunsky coefficients obeying a Coulomb-type decay estimate have no singular continuous component.

We introduce the boolean convolution for probability measures on the unit circle. Roughly speaking, it describes the distribution of the product of two boolean independent unitary random variables. We find an analogue of the characteristic…

Let $\mu$ be a Borel probability measure on $\mathrm{SL}_2(\mathbb R)$ with a finite exponential moment, and assume that the subgroup $\Gamma_{\mu}$ generated by the support of $\mu$ is Zariski dense. Let $\nu$ be the unique…

It is shown that \begin{equation*} \mathsf{P}(\|a_1U_1+\dots+a_nU_n\|>u)\le c\,\mathsf{P}(a\|Z_d\|>u) \end{equation*} for all real $u$, where $U_1,\dots,U_n$ are independent random vectors uniformly distributed on the unit sphere in…

Solomonoff's central result on induction is that the posterior of a universal semimeasure M converges rapidly and with probability 1 to the true sequence generating posterior mu, if the latter is computable. Hence, M is eligible as a…

This paper contributes to the study of the free additive convolution of probability measures. It shows that under some conditions, if measures $\mu_i$ and $\nu_i, i=1,2$, are close to each other in terms of the L\'{e}vy metric and if the…

One of the main concepts in quantum physics is a density matrix, which is a symmetric positive definite matrix of trace one. Finite probability distributions are a special case where the density matrix is restricted to be diagonal. Density…

In this paper we prove a central limit theorem for some probability measures defined as asymtotic densities of integer sets defined via sum-of-digit-function. To any integer a we can associate a measure on Z called $\mu$a such that, for any…

We investigate random Bernoulli convolutions, namely, probability measures given by the infinite convolution \[ \mu_\omega = \mathop{\circledast}_{k=1}^{\infty} \left( \frac{\delta_0 + \delta_{\lambda_1 \lambda_2 \ldots \lambda_{k-1}…