Independent Uniform Random Variables Research Articles

Random matrices associated to Young diagrams

We consider the singular values of certain Young diagram shaped random matrices. For block-shaped random matrices, the empirical distribution of the squares of the singular eigenvalues converges almost surely to a distribution whose moments are a generalization of the Catalan numbers. The limiting distribution is the density of a product of rescaled independent Beta random variables and its Stieltjes–Cauchy transform has a hypergeometric representation. In special cases we recover the Marchenko–Pastur and Dykema–Haagerup measures of square and triangular random matrices, respectively. We find a further factorization of the moments in terms of two complex-valued random variables that generalizes the factorization of the Marchenko–Pastur law as product of independent uniform and arcsine random variables.

Discrete Optimal Transport with Independent Marginals is #P-Hard

We study the computational complexity of the optimal transport problem that evaluates the Wasserstein distance between the distributions of two -dimensional discrete random vectors. The best known algorithms for this problem run in polynomial time in the maximum of the number of atoms of the two distributions. However, if the components of either random vector are independent, then this number can be exponential in even though the size of the problem description scales linearly with . We prove that the described optimal transport problem is #P-hard even if all components of the first random vector are independent uniform Bernoulli random variables, while the second random vector has merely two atoms, and even if only approximate solutions are sought. We also develop a dynamic programming–type algorithm that approximates the Wasserstein distance in pseudo-polynomial time when the components of the first random vector follow arbitrary independent discrete distributions, and we identify special problem instances that can be solved exactly in strongly polynomial time.

Sharp bounds on p-norms for sums of independent uniform random variables, 0 < p < 1

Sharp bounds on p-norms for sums of independent uniform random variables, 0 < p < 1

Dirac distributions related to sums of independent nonidentically uniform random variables

The aim of this note is to give an elegant proof of a result due to E. G. Olds which concerns the density distribution of the sum of independent uniform random variables non-identically distributed. The proof uses both analytical and combinatorial properties of Dirac distributions and their convolutions. The method is new and can apply to other situations.

From Ball's cube slicing inequality to Khinchin-type inequalities for negative moments

We establish a sharp moment comparison inequality between an arbitrary negative moment and the second moment for sums of independent uniform random variables, which extends Ball's cube slicing inequality.

Teaching Random Variables

This paper presents demo program for teaching random variables dedicated to demonstration and visualization of behaviour of the product of m independent uniform random variables. The demo is used as a complementary tool in the teaching Random signals and processes curriculum at the graduate level. The demo is made in MATLAB and utilizes all benefits provided by MATLAB, such as simplicity, and good graphic environment, interactive use, among others. Additionally, this demo is used with other demo programs, also made in MATLAB, in teaching/learning process of random variables. Students were very satisfied with this demo program expressing that helped them understanding better the behaviour of the product of uniform random variables.

An improved lower bound for the Traveling Salesman constant

Let X1,X2,…,Xn be independent uniform random variables on [0,1]2. Let L(X1,…,Xn) be the length of the shortest Traveling Salesman tour through these points. Beardwood et al (1959) showed that there exists a constant β such that limn→∞L(X1,…,Xn)n=βalmost surely. It was shown that β≥0.625. Building upon an approach proposed by Steinerberger (2015), we improve the lower bound to β≥0.6277.

Inverting Khintchine’s relationship and generating length biased data

If X>0 follows a distribution with decreasing density, then Khintchine’s theorem states that it has the same distribution as U×S where U and S>0 are independent, U following the uniform distribution on (0,1). In this letter, an explicit function of X and independent V∼U(0,1) is discovered which has the same distribution as S. This result is then used to find an explicit function of two independent uniform random variables which follows the length biased form of a general distribution on R+ with finite mean.

Multi-Panel Kendall plot in light of an ROC curve analysis applied to measuring dependence

ABSTRACTThe Kendall plot (-plot) is a plot measuring dependence between the components of a bivariate random variable. The -plot graphs the Kendall distribution function against the distribution function of VU, where V and U are independent uniform random variables. We associate -plots with the receiver operating characteristic () curve, a well-accepted graphical tool in biostatistics for evaluating the ability of a biomarker to discriminate between two populations. The most commonly used global index of diagnostic accuracy of biomarkers is the area under the curve (). In parallel with the , we propose a novel strategy to measure association between random variables from a continuous bivariate distribution. First, we discuss why the area under the conventional Kendall curve () cannot be used as an index of dependence. We then suggest a simple and meaningful extension of the definition of the -plots, and define an index of dependence that is based on . This measure characterizes a wide range of two-variable relationships, thereby completely detecting the underlying dependence structure. Properties of the proposed index satisfy the mathematical definition of a measure. Finally, simulations and real data examples illustrate the applicability of the proposed method.

A PDE approach to a 2-dimensional matching problem

We prove asymptotic results for 2-dimensional random matching problems. In particular, we obtain the leading term in the asymptotic expansion of the expected quadratic transportation cost for empirical measures of two samples of independent uniform random variables in the square. Our technique is based on a rigorous formulation of the challenging PDE ansatz by Caracciolo et al. (Phys Rev E 90:012118, 2014) that linearizes the Monge–Ampère equation.

Evaluation of harmonic sums with integrals

We consider the sums $S(k)=\sum_{n=0}^{\infty}\frac{(-1)^{nk}}{(2n+1)^k}$ and $\zeta(2k)=\sum_{n=1}^{\infty}\frac{1}{n^{2k}}$ with $k$ being a positive integer. We evaluate these sums with multiple integration, a modern technique. First, we start with three different double integrals that have been previously used in the literature to show $S(2)=\pi^2/8,$ which implies Euler's identity $\zeta(2)=\pi^2/6.$ Then, we generalize each integral in order to find the considered sums. The $k$ dimensional analogue of the first integral is the density function of the quotient of $k$ independent, nonnegative Cauchy random variables. In seeking this function, we encounter a special logarithmic integral that we can directly relate to $S(k).$ The $k$ dimensional analogue of the second integral, upon a change of variables, is the volume of a convex polytope, which can be expressed as a probability involving certain pairwise sums of $k$ independent uniform random variables. We use combinatorial arguments to find the volume, which in turn gives new closed formulas for $S(k)$ and $\zeta(2k).$ The $k$ dimensional analogue of the last integral, upon another change of variables, is an integral of the joint density function of $k$ Cauchy random variables over a hyperbolic polytope. This integral can be expressed as a probability involving certain pairwise products of these random variables, and it is equal to the probability from the second generalization. Thus, we specifically highlight the similarities in the combinatorial arguments between the second and third generalizations.

Online purchasing under uncertainty

AbstractSuppose there is a collection of independent uniform random variables, and a hypergraph of target structures on the vertex set . We would like to purchase a target structure at small cost, but we do not know all the costs xi ahead of time. Instead, we inspect the random variables xi one at a time, and after each inspection, choose to either keep the vertex i at cost xi, or reject vertex i forever. In the present paper, we consider the case where is the edge‐set of a complete graph (or digraph), and the target structures are the spanning trees of a graph, spanning arborescences of a digraph, the paths between a fixed pair of vertices, perfect matchings, Hamilton cycles or the cliques of some fixed size.

Sum of squares of uniform random variables

Given a set of n independent uniform random variables on [0,1], this paper deals with the distribution of their sum of squares. Explicit solutions are given for n=2,3 and 4. Graphical presentations are given for n up to 12. The case n=2 is special in the sense that the density function is constant on [0,1], a property noticed first by Adi Ben-Israel.

A Geometric Derivation of the Irwin-Hall Distribution

The Irwin-Hall distribution is the distribution of the sum of a finite number of independent identically distributed uniform random variables on the unit interval. Many applications arise since round-off errors have a transformed Irwin-Hall distribution and the distribution supplies spline approximations to normal distributions. We review some of the distribution’s history. The present derivation is very transparent, since it is geometric and explicitly uses the inclusion-exclusion principle. In certain special cases, the derivation can be extended to linear combinations of independent uniform random variables on other intervals of finite length. The derivation adds to the literature about methodologies for finding distributions of sums of random variables, especially distributions that have domains with boundaries so that the inclusion-exclusion principle might be employed.

Exponential–uniform identities related to records

Symmetry properties of the random grid spanned on the set of records from the planar Poisson point process imply bizarre multivariate distributional identities for certain rational functions of independent exponential and uniform random variables.

On the mean residual life order of convolutions of independent uniform random variables

Let Xλi,i=1,…,n be n independent random variables such that Xλi has uniform distribution over the interval (0,1/λi),i=1,…,n. It is proved that if (λ1,…,λn) is larger than (λ1⁎,…,λn⁎) according to reciprocal order, then ∑i=1nXλi is larger than ∑i=1nXλi⁎ according to mean residual life order as well as increasing convex order. This result gives convenient bounds for mean residual life function of ∑i=1nXλi in terms of harmonic mean of λi's. It is shown that these bounds are sharper than those given in the literature in terms of geometric mean and arithmetic mean of λi's.

The distribution of the product of powers of independent uniform random variables — A simple but useful tool to address and better understand the structure of some distributions

What is the distribution of the product of given powers of independent uniform (0, 1) random variables? Is this distribution useful? Is this distribution commonly used in some contexts? Is this distribution somehow related to the distribution of the product of other random variables? Are there some test statistics with this distribution? This paper will give the answers to the above questions. It will be seen that the answer to the last four questions above is: yes! We will show how particular choices of the numbers of variables involved and their powers will result in interesting and useful distributions and how these distributions may help us to shed some new light on some well-known distributions and also how it may help us to address, in a much simpler way, some distributions usually considered to be rather complicated as is the case with the exact distribution of a number of statistics used in Multivariate Analysis, including some whose exact distribution up until now is not available in a concise and manageable form.

Perfect matchings as IID factors on non-amenable groups

We prove that in every bipartite Cayley graph of every non-amenable group, there is a perfect matching that is obtained as a factor of independent uniform random variables. We also discuss expansion properties of factors and improve the Hoffman spectral bound on the independence number of finite graphs.

A Survey on Simulating Stable Random Variables

In general case, Chambers et al. (1976) introduced the following algorithm for simulating any stable random variables $ X/sim(alpha, beta, gamma, delta) $ with four parameters. They use a nonlinear transformation of two independent uniform random variables for simulating an stable random variable... (to continue, click here)

Truncated stable random variables: characterization and simulation

Kanter (Ann Probab 3(4):697–707, 1975) and Chambers et al. (J Am Stat Assoc 71(354):340–344, 1976) developed a method for characterizing and simulating stable random variables, X, using nonlinear transformations involving two independent uniform random variables. Their method is scrutinized to provide a characterization and then develop a method for simulating random variables with distribution P(X ≤ x| X > a), called here truncated stable random variables. Our characterization is rigorous when the characteristic exponent α ≠ 1. We extend our method to the case that α → 1.