Related papers: The uniform distribution of sequences generated by…

Building on the work of Arizmendi and Celestino (2021), we derive the $*$-distributions of polynomials in monotone independent and infinitesimally monotone independent elements. For non-zero complex numbers $\alpha$ and $\beta$, we derive…

Let $z_1, \dots, z_m$ be $m$ distinct complex numbers, normalized to $|z_k| = 1$, and consider the polynomial $$ p_{m}(z) = \prod_{k=1}^{m}{(z-z_k)}.$$ We define a sequence of polynomials in a greedy fashion, $$ p_{N+1}(z) = p_{N}(z)…

One of the fundamental theorems of uniform distribution theory states that the fractional parts of the sequence $(n \alpha)_{n \geq 1}$ are uniformly distributed modulo one (u.d. mod 1) for every irrational number $\alpha$. Another…

Random permutations with distribution conditionally uniform given the set of record values can be generated in a unified way, coherently for all values of $n$. Our central example is a two-parameter family of random permutations that are…

We derive a necessary and sufficient condition for the sum of M independent continuous random variables modulo 1 to converge to the uniform distribution in L^1([0,1]), and discuss generalizations to discrete random variables. A consequence…

We investigate the probability that a random polynomial with independent, mean-zero and finite variance coefficients has no real zeros. Specifically, we consider a random polynomial of degree $2n$ with coefficients given by an i.i.d.…

Choose n random, independent points in R^d according to a fixed distribution. The convex hull of these points is a random polytope. In some cases, central limit theorems have been proven for the components of f-vectors of random polytopes…

Let $\Mmm$ denote the set of $\mm$ matrices with complex entries, and let $\calG(\partial_1,...,\partial_n)$ be an $\mm$ matrix whose entries are partial differential operators on $\Rn$ with constant complex coefficients. It is proved that…

In this paper, we study the distribution of the cokernels of random $p$-adic matrices with fixed zero entries. Let $X_n$ be a random $n \times n$ matrix over $\mathbb{Z}_p$ in which some entries are fixed to be zero and the other entries…

We prove new statistical results about the distribution of the cokernel of a random integral matrix with a concentrated residue. Given a prime $p$ and a positive integer $n$, consider a random $n \times n$ matrix $X_n$ over the ring…

In a recent work Killip and Nenciu gave random recurrences for the characteristic polynomials of certain unitary and real orthogonal upper Hessenberg matrices. The corresponding eigenvalue p.d.f.'s are beta-generalizations of the classical…

A numerical semigroup is a subset of the non-negative integers that is closed under addition. For a randomly generated numerical semigroup, the expected number of minimum generators can be expressed in terms of a doubly-indexed sequence of…

Let $M$ be a random matrix chosen according to Haar measure from the unitary group $\mathrm{U}(n,\mathbb{C})$. Diaconis and Shahshahani proved that the traces of $M,M^2,\ldots,M^k$ converge in distribution to independent normal variables as…

Two closely related discrete probability distributions are introduced. In each case the support is a set of vectors in $\mathbb{R}^n$ obtained from the partitions of the fixed positive integer $n$. These distributions arise naturally when…

Fix a modulus $q$. One would expect the number of primes in each invertible residue class mod $q$ to be multinomially distributed, i.e. for each $p \,\mathrm{mod}\, q$ to behave like an independent random variable uniform on…

Given an increasing sequence of integers a(n), it is known (due to Weyl) that for almost all reals t, the fractional parts of the dilated sequence t*a(n) are uniformly distributed in the unit interval. Some effort has been made recently to…

For a given monic integral polynomial $f(x)$ of degree $n$, we define local roots $r_i$ of $f(x)$ for a completely decomposable prime $p$ by $r_i \in \mathbb{Z}$, $f(r_i) \equiv 0 \bmod p$ and $0 \le r_1 \le r_2 \le \dots \le r_n < p$. With…

Let $\Theta^{(n)}$ be a random vector uniformly distributed on the unit sphere $\mathbb S^{n-1}$ in $\mathbb R^n$. Consider the projection of the uniform distribution on the cube $[-1,1]^n$ to the line spanned by $\Theta^{(n)}$. The…

The Kolakoski sequence is the unique infinite sequence with values in $\{1,2\}$ and first term $1$ which equals the sequence of run-lengths of itself, we call this $K(1,2).$ We define $K(m,n)$ similarly. A well-known conjecture is that the…

Let \( \{\varphi_i\}_{i=0}^\infty \) be a sequence of orthonormal polynomials on the unit circle with respect to a probability measure \( \mu \). We study zero distribution of random linear combinations of the form \[…