Related papers: The Complex Roots of Random Sums

The average density of zeros for monic generalized polynomials, $P_n(z)=\phi(z)+\sum_{k=1}^nc_kf_k(z)$, with real holomorphic $\phi ,f_k$ and real Gaussian coefficients is expressed in terms of correlation functions of the values of the…

Many statistics of roots of random polynomials have been studied in the literature, but not much is known on the concentration aspect. In this note we present a systematic study of this question, aiming towards nearly optimal bounds to some…

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…

Let $\{\eta_{j}\}_{j = 0}^{N}$ be a sequence of independent, identically distributed random complex Gaussian variables, and let $\{f_{j} (z)\}_{j = 0}^{N}$ be a sequence of given analytic functions that are real-valued on the real number…

We derive a simple and precise approximation to probability density functions in sampling distributions based on the Fourier cosine series. After clarifying the required conditions, we illustrate the approximation on two examples: the…

The distribution of the sum of r-th power of standard normal random variables is a generalization of the chi-squared distribution. In this paper, we represent the probability density function of the random variable by an one-dimensional…

In this paper an analytic expression is given for the bounds of the distribution function of the sum of dependent normally distributed random variables. Using the theory of copulas and the important Frechet bounds the dependence structure…

We study the asymptotic distribution of roots of Lommel polynomials as polynomials of the order with a variable and purely imaginary argument. The roots are complex and accumulate on certain curves in the complex plane. We prove existence…

The notion of probability density for a random function is not as straightforward as in finite-dimensional cases. While a probability density function generally does not exist for functional data, we show that it is possible to develop the…

We consider the zeros of the sum of independent random polynomials as their degrees tend to infinity. Namely, let $p$ and $q$ be two independent random polynomials of degree $n$, whose roots are chosen independently from the probability…

We seek random versions of some classical theorems on complex approximation by polynomials and rational functions, as well as investigate properties of random compact sets in connection to complex approximation.

We give formulas for the number of polynomials over a finite field with given root multiplicities, in particular in cases when the formula is surprisingly simple (a power of q). Besides this concrete interpretation, we also prove an…

We generalize some previous results on random polynomials in several complex variables. A standard setting is to consider random polynomials $H_n(z):=\sum_{j=1}^{m_n} a_jp_j(z)$ that are linear combinations of basis polynomials $\{p_j\}$…

For each $\alpha \in (0, 1)$, we construct a bounded monotone deterministic sequence $(c_k)_{k \geq 0}$ of real numbers so that the number of real roots of the random polynomial $f_n(z) = \sum_{k=0}^n c_k \varepsilon_k z^k$ is $n^{\alpha +…

Polynomials are common algebraic structures, which are often used to approximate functions including probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of…

We study the regularity of densities of distributions that are polynomial images of the standard Gaussian measure on $\mathbb{R}^n$. We assume that the degree of a polynomial is fixed and that each variable enters to a power bounded by…

The paper addresses the calculation of correlation functions of permanental polynomials of matrices with random entries. By exploiting a convenient contour integral representation of the matrix permanent some explicit results are provided…

We establish a normal approximation for the limiting distribution of partial sums of random Rademacher multiplicative functions over function fields, provided the number of irreducible factors of the polynomials is small enough. This…

This work is divided into three parts. The first part concerns polynomials in one variable with all real roots. We consider linear transformations that preserve real rootedness, as well as matrices that preserve interlacing. The second part…

In the paper I study properties of random polynomials with respect to a general system of functions. Some lower bounds for the mathematical expectation of the uniform and recently introduced integral-uniform norms of random polynomials are…