Related papers: Supremum of Random Dirichlet Polynomials with Sub-…

Let $B_{n}(t)$ be a $n$-th Stern polynomial and let $e(n)=\op{deg}B_{n}(t)$ be its degree. In this note we continue our study started in \cite{Ul} of the arithmetic properties of the sequence of Stern polynomials and the sequence…

We attach a certain $n \times n$ matrix $A_n$ to the Dirichlet series $L(s)=\sum_{k=1}^{\infty}a_k k^{-s}$. We study the determinant, characteristic polynomial, eigenvalues, and eigenvectors of these matrices. The determinant of $A_n$ can…

Consider the divisor sum $\sum_{n\leq N}\tau(n^2+2bn+c)$ for integers $b$ and $c$. We extract an asymptotic formula for the average divisor sum in a convenient form, and provide an explicit upper bound for this sum with the correct main…

Directional data require specialized probability models because of the non-Euclidean and periodic nature of their domain. When a directional variable is observed jointly with linear variables, modeling their dependence adds an additional…

We consider an infinitely divisible random field indexed by $\mathbb{R}^d$, $d\in\mathbb{N}$, given as an integral of a kernel function with respect to a L\'evy basis with a L\'evy measure having a regularly varying right tail. First we…

We investigate the approximation for computing the sum $a_1+...+a_n$ with an input of a list of nonnegative elements $a_1,..., a_n$. If all elements are in the range $[0,1]$, there is a randomized algorithm that can compute an…

In the maximum independent set of convex polygons problem, we are given a set of $n$ convex polygons in the plane with the objective of selecting a maximum cardinality subset of non-overlapping polygons. Here we study a special case of the…

This paper derives a new strong Gaussian approximation bound for the sum of independent random vectors. The approach relies on the optimal transport theory and yields \textit{explicit} dependence on the dimension size $p$ and the sample…

Let $\mathcal{F}$ be a set consisting of $n$ real vectors of dimension $m \leq n$. For any saturated, i.e., $m$-element, subset $\mathcal{S}$ of $\mathcal{F}$, let $\mathrm{vol}(\mathcal{S})$ be the volume of the parallelotope formed by the…

We study a two-dimensional discrete directional maximal operator along the set of the prime numbers. We show existence of a set of vectors, which are lattice points in a sufficiently large annulus, for which the $\ell^2$ norm of the…

We present a novel distribution-free approach, the data-driven threshold machine (DTM), for a fundamental problem at the core of many learning tasks: choose a threshold for a given pre-specified level that bounds the tail probability of the…

An interesting line of research is the investigation of the laws of random variables known as Dirichlet means. However, there is not much information on interrelationships between different Dirichlet means. Here, we introduce two…

We consider the random continued fraction S(t) := 1/(s_1 + t/(s_2 + t/(s_3 + >...))) where the s_n are independent random variables with the same gamma distribution. For every realisation of the sequence, S(t) defines a Stieltjes function.…

Evaluating a polynomial on a set of points is a fundamental task in computer algebra. In this work, we revisit a particular variant called trimmed multipoint evaluation: given an $n$-variate polynomial with bounded individual degree $d$ and…

We study the upper tail of the number of arithmetic progressions of a given length in a random subset of {1,...,n}, establishing exponential bounds which are best possible up to constant factors in the exponent. The proof also extends to…

We consider the problem of drawing samples from posterior distributions formed under a Dirichlet prior and a truncated multinomial likelihood, by which we mean a Multinomial likelihood function where we condition on one or more counts being…

Let $M$ be an $n\times n$ random matrix with entries in $\{0, 1\}$, where each row is independently and uniformly sampled from the set of all vectors in $\{0, 1\}^n$ containing exactly $d$ ones, with $d=pn$ for some fixed constant $p\in…

We obtain asymptotic for the quantity $\int_0^1 \bigg|\sum_{n\le X}\tau_k(n)e(n\alpha)\bigg|d\alpha$ where $\tau_k(n) = \sum_{d_1\dots d_k = n} 1$. This follows from a quick application of the circle method. Along the way, we find minor arc…

Integrating probability and nonprobability survey samples is an important problem in modern survey sampling. Nonprobability samples often contain rich outcome information but may lack population representativeness, whereas probability…

Let $M_n$ be the maximum of $n$ zero-mean gaussian variables $X_1,..,X_n$ with covariance matrix of minimum eigenvalue $\lambda$ and maximum eigenvalue $\Lambda$. Then, for $n \ge 70$, $$\Pr\{M_n \ge \lambda \left (2 \log n - 2.5 - \log(2…