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In the context of phase-space quantization, matrix elements and observables result from integration of c-number functions over phase space, with Wigner functions serving as the quasi-probability measure. The complete sets of Wigner…

The uniform probability measure on a convex polytope induces piecewise polynomial densities on its projections. For a fixed combinatorial type of simplicial polytopes, the moments of these measures are rational functions in the vertex…

Seminal works by Birch and Ihara gave formulas for the $m$th power moments of the traces of Frobenius endomorphisms of elliptic curves over $\mathbb{F}_{p}$ for primes $p \geq 5$. Recent works by Kaplan and Petrow generalized these results…

Let $X_1,\ldots,X_n$ be independent random points that are distributed according to a probability measure on $\mathbb{R}^d$ and let $P_n$ be the random convex hull generated by $X_1,\ldots,X_n$ ($n\geq d+1$). Natural classes of probability…

This paper investigates the usage of generating functions (GFs) encoding measures over the program variables for reasoning about discrete probabilistic programs. To that end, we define a denotational GF-transformer semantics for…

Let $f$ be a positive multiplicative function and let $k\geq 2$ be an integer. We prove that if the prime values $f(p)$ converge to $1$ sufficiently slowly as $p\rightarrow +\infty$, in the sense that $\sum_{p}|f(p)-1|=\infty$, there exists…

We examine two different ways of encoding a counting function, as a rational generating function and explicitly as a function (defined piecewise using the greatest integer function). We prove that, if the degree and number of input…

We investigate the joint moments of the 2k-th power of the characteristic polynomial of random unitary matrices with the 2h-th power of the derivative of this same polynomial. We prove that for a fixed h, the moments are given by rational…

In this paper we introduce a biparametric family of transformations which can be seen as an extension of the so-called up and down transformations. This new class of transformations allows to us to introduce new informational functionals,…

Network calculus is a min-plus system theory for performance evaluation of queuing networks. Its elegance stems from intuitive convolution formulas for concatenation of deterministic servers. Recent research dispenses with the worst-case…

We discuss conditions under which a convex cone $\K\subset \R^{\Omega}$ admits a probability $m$ such that $\sup_{k\in \K} m(k)\leq0$. Based on these, we also characterize linear functionals that admit the representation as finitely…

Recently, Banderier et. al. considered Young tableaux with walls, which are similar to standard Young tableaux, except that local decreases are allowed at some walls. We count the numbers $\overline{f}_m(n)$ of Young tableaux of shape…

We study the Fourier spectrum of functions $f\colon \{0,1\}^{mk} \to \{-1,0,1\}$ which can be written as a product of $k$ Boolean functions $f_i$ on disjoint $m$-bit inputs. We prove that for every positive integer $d$, \[ \sum_{S \subseteq…

In this paper, we introduce a new class of polynomials, called probabilistic q-Bernstein polynomials, alongside their generating function. Assuming Y is a random variable satisfying moment conditions, we use the generating function of these…

A set of necessary and sufficient conditions for a sequence of moment generating functions to converge to a moment generating function on an interval (a,b) not necessarily containing 0, is given. The result is derived using recent results…

A unifying and generalizing approach to representations of the positive-part and absolute moments $\mathsf{E} X_+^p$ and $\mathsf{E}|X|^p$ of a random variable $X$ for real $p$ in terms of the characteristic function (c.f.) of $X$, as well…

Hirschman and Widder introduced a class of P\'olya frequency functions given by linear combinations of one-sided exponential functions. The members of this class are probability densities, and the class is closed under convolution but not…

In this paper, we generalize and improve some fundamental concentration inequalities using information on the random variables' higher moments. In particular, we improve the classical Hoeffding's and Bennett's inequalities for the case…

There has been significant interest in studying the asymptotics of certain generalised moments, called the moments of moments, of characteristic polynomials of random Haar-distributed unitary and symplectic matrices, as the matrix size $N$…

We derive a simple expression for the $r^{th}$ factorial moment $\mu_{(r)}$ of the geometric distribution of order $k$ with success parameter $p\in(0,1)$ (and $q=1-p$) in terms of its probability mass function $f_k(n)$. Specifically,…