On the number of particles from a marked set of cells for an analogue of a general allocation scheme

Consider a sequence of polynomials of bounded degree evaluated in independent Gaussian, Gamma or Beta random variables. We show that, if this sequence converges in law to a nonconstant distribution, then (1) the limit distribution is necessarily absolutely continuous with respect to the Lebesgue measure and (2) the convergence automatically takes place in the total variation topology. Our proof, which relies on the Carbery–Wright inequality and makes use of a diffusive Markov operator approach, extends the results of Nourdin and Poly (Stoch Proc Appl 123:651–674, 2013) to the Gamma and Beta cases.

CHAPTER 4 - The Number and Expected Number of Real Zeros of Random Algebraic Polynomials

There are many previous studies on designing efficient and high-order numerical methods for stochastic differential equations (SDEs) driven by Gaussian random variables. They mostly focus on proposing numerical methods for SDEs with independent Gaussian random variables and rarely solving SDEs driven by dependent Gaussian random variables. In this paper, we propose a Galerkin spectral method for solving SDEs with dependent Gaussian random variables. Our main techniques are as follows: (1) We design a mapping transformation between multivariate Gaussian random variables and independent Gaussian random variables based on the covariance matrix of multivariate Gaussian random variables. (2) First, we assume the unknown function in the SDE has the generalized polynomial chaos expansion and convert it to be driven by independent Gaussian random variables by the mapping transformation; second, we implement the stochastic Galerkin spectral method for the SDE in the Gaussian measure space; and third, we obtain deterministic differential equations for the coefficients of the expansion. (3) We employ a spectral method solving the deterministic differential equations numerically. We apply the newly proposed numerical method to solve the one-dimensional and two-dimensional stochastic Poisson equations and one-dimensional and two-dimensional stochastic heat equations, respectively. All the presented stochastic equations are driven by two Gaussian random variables, and they are dependent and have multivariate normal distribution of their joint probability density.

Determining distributions of the functions of random variables is a very important problem with a wide range of applications in Risk Management, Finance, Economics, Science, and many other areas. This paper develops the theory on both density and distribution functions for the quotient Y = X 1 X 2 and the ratio of one variable over the sum of two variables Z = X 1 X 1 + X 2 of two dependent or independent random variables X 1 and X 2 by using copulas to capture the structures between X 1 and X 2 . Thereafter, we extend the theory by establishing the density and distribution functions for the quotients Y = X 1 X 2 and Z = X 1 X 1 + X 2 of two dependent normal random variables X 1 and X 2 in the case of Gaussian copulas. We then develop the theory on the median for the ratios of both Y and Z on two normal random variables X 1 and X 2 . Furthermore, we extend the result of median for Z to a larger family of symmetric distributions and symmetric copulas of X 1 and X 2 . Our results are the foundation of any further study that relies on the density and cumulative probability functions of ratios for two dependent or independent random variables. Since the densities and distributions of the ratios of both Y and Z are in terms of integrals and are very complicated, their exact forms cannot be obtained. To circumvent the difficulty, this paper introduces the Monte Carlo algorithm, numerical analysis, and graphical approach to efficiently compute the complicated integrals and study the behaviors of density and distribution. We illustrate our proposed approaches by using a simulation study with ratios of normal random variables on several different copulas, including Gaussian, Student-t, Clayton, Gumbel, Frank, and Joe Copulas. We find that copulas make big impacts from different Copulas on behavior of distributions, especially on median, spread, scale and skewness effects. In addition, we also discuss the behaviors via all copulas above with the same Kendall’s coefficient. The approaches developed in this paper are flexible and have a wide range of applications for both symmetric and non-symmetric distributions and also for both skewed and non-skewed copulas with absolutely continuous random variables that could contain a negative range, for instance, generalized skewed-t distribution and skewed-t Copulas. Thus, our findings are useful for academics, practitioners, and policy makers.

Hamburger moment problem for powers and products of random variables

In this paper, we extend Stein’s method to products of independent beta, gamma, generalised gamma and mean zero normal random variables. In particular, we obtain Stein operators for mixed products of these distributions, which include the classical beta, gamma and normal Stein operators as special cases. These operators lead us to closed-form expressions involving the Meijer $G$-function for the probability density function and characteristic function of the mixed product of independent beta, gamma and central normal random variables.

Probability and Random Processes for Electrical Engineering

We address the problem of robust inference about the stress–strength reliability parameter R = P(X < Y), where X and Y are taken to be independent random variables. Indeed, although classical likelihood based procedures for inference on R are available, it is well-known that they can be badly affected by mild departures from model assumptions, regarding both stress and strength data. The proposed robust method relies on the theory of bounded influence M-estimators. We obtain large-sample test statistics with the standard asymptotic distribution by means of delta-method asymptotics. The finite sample behavior of these tests is investigated by some numerical studies, when both X and Y are independent exponential or normal random variables. An illustrative application in a regression setting is also discussed.

The ratio of independent random variables arises in many applied problems. The distribution of the ratio |X/Y| is studied when X and Y are independent Normal and Rice random variables, respectively. Ratios of such random variables have extensive applications in the analysis of noises in communication systems. The exact forms of probability density function (PDF), cumulative distribution function (CDF) and the existing moments are derived in terms of several special functions. As a special case, the PDF and CDF of the ratio of independent standard Normal and Rayleigh random variables have been obtained. Tabulations of associated percentage points and a computer program for generating tabulations are also given.

In this letter, we derive new results for the statistics of the complex random variable <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</i> = <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Delta</sup> Sigma <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n=1</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sup> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> = <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> + <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">jz</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</sub> = <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">re</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">jphi</sup> where { <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> } is a set of mutually independent complex-valued Gaussian random variables with arbitrary means and equal variances. Each random variable <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> is assumed to have independent real and imaginary components with equal variance for all <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> . Expressions are derived for the joint probability density function (pdf) of ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</sub> ), for the joint pdf of ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> ,phi) and also for the marginal pdf of the modulus <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</i> . An useful Fourier series expansion for the pdf of the phase phi is also derived. As an application of the results, a theoretical performance analysis of the well-known nondata-aided Viterbi and Viterbi feedforward carrier phase estimator operating with BPSK signals is presented. In particular, the expressions for the exact pdf, variance, and equivocation probability of the carrier phase estimates are derived.

Cycle-transitive comparison of independent random variables

Researchers commonly construct histograms as a first step in representing and visualizing their geospatial data. Because of the presence of spatial autocorrelation in these data, these graphs usually fail to closely align with any of the several hundred existing ideal frequency distributions. The purpose of this paper is to address how positive spatial autocorrelation — the most frequently encountered in practice — can distort histograms constructed with geospatial data. Following the auto-normal parameter specification employed in WinBUGS for Bayesian analysis, this paper summarizes results for normal, Poisson, and binomial random variables (RVs) — three of the most commonly employed ones by geospatial scientists — in terms of mixture distributions. A spatial filter description of positive spatial autocorrelation is shown to approximate a normal distribution in its initial form, a gamma distribution when exponentiated, and a beta distribution when embedded in a logistic equation. In turn, these conceptualizations allow: the mean for a normal distribution to be distributed as a normal random variable (RV) with a zero mean and a specific variance; the mean for a Poisson distribution to be distributed as a gamma RV with specific parameters (i.e., a negative binomial distribution); and, the probability for a binomial distribution to be distributed as a beta RV with specific parameters (i.e., a beta-binomial distribution). Results allow impacts of positive spatial autocorrelation on histograms to be better understood. A methodology is outlined for recovering the underlying unautocorrelated frequency distributions.

Real-Time Kinematic GPS positioning is widely used for many applications.Resolving ambiguities is the key to precise positioning. Integer ambiguity resolution isthe process of resolving the unknown cycle ambiguities of double difference carrierphase data as integers. Two important issues of resolving are efficiency andreliability. In the conventional search techniques, we generally used chi-squarerandom variables for decision variables. Mathematically, a chi-square random variableis the sum of mutually independent, squared zero-mean unit-variance normal(Gaussian) random variables. With this base knowledge, we can separate decisionvariables to several normal random variables. We showed it with related equationsand conceptual diagrams. With this separation, we can improve the computationalefficiency of the process without losing the needed performance. If we averageseparated normal random variables sequentially, averaged values are also normalrandom variables. So we can use them as decision variables, which prevent from asudden increase of some decision variable. With the method using averaged decisionvalues, we can get the solution more quicklv and more reliably.To verify the performance of our proposed algorithm, we conducted simulations.We used some visual diagrams that are useful for intuitional approach. We analyzedthe performance of the proposed algorithm and compared it to the conventionalmethods.

Determining distributions of the functions of random variables is a very important problem with wide applications in Risk Management, Finance, Economics, Science, and many other areas. This paper develops the theory on both density and distribution functions for the quotient Y = X1/X 2 and the ratio of one variable over the sum of two variables Z = X1/X1+X2 of two dependent or independent random variables X1 and X2 by using copulas to capture the structures between X1 and X2. We then extend the theory by establishing the density and distribution functions for the quotients Y = X1/ X2 and Z = X1/X1+X2 of two dependent normal random variables X1 and X2 in case of Gaussian copulas. Thereafter, we develop the theory on the median for the ratios of both Y and Z on two normal random variables X1 and X2. Furthermore, the result of median for Z is also extended to a larger family of symmetric distributions and symmetric copulas of X1 and X2. Our results are the foundation of any further study that relies on the density and cumulative probability functions of ratios for two dependent random variables. Since the densities and distributions of the ratios of both Y and Z are in terms of integrals and are very complicated, their exact forms cannot be obtained. To circumvent the difficulty, this paper introduces the Monte Carlo algorithm, numerical analysis, and graphical approach to efficiently compute the complicated integrals and study the behaviors of density and distribution. We illustrate our proposed approaches by using a simulation study with ratios of normal random variables on several different copulas, including Gaussian, Student-t, Clayton, Gumbel, Frank, and Joe Copulas. We find that copulas make big impacts from different Copulas on behavior of distributions, especially on median, spread, scale and skewness effects. In addition, we also discuss the behaviors via all copulas above with the same Kendall's coefficient. Our findings are useful for academics, practitioners, and policy makers.

Given n (discrete or continuous) random variables Xi, the (2n - 1)-dimensional vector obtained by evaluating the joint entropy of all non-empty subsets of {X1,hellip, Xn} is called an entropic vector. Determining the region of entropic vectors is an important open problem in information theory. Recently, Chan has shown that the entropy regions for discrete and continuous random variables, though different, can be determined from one another. An important class of continuous random variables are those that are vector-valued and jointly Gaussian. It is known that Gaussian random variables violate the Ingleton bound, which many random variables such as those obtained from linear codes over finite fields do satisfy, and they also achieve certain non-Shannon inequalities. In this paper we give a full characterization of the entropy region for three jointly-Gaussian vector-valued random variables and, rather surprisingly, show that the region is strictly smaller than the entropy region for three arbitrary random variables. However, we also show the following result. For any given entropic vector h isin R7, there exists a thetas* > 0, such that for all thetas ges thetas*, the vector 1/thetas h can be generated by three vector-valued jointly Gaussian random variables. This implies that for three random variables the region of entropic vectors can be obtained by considering the cone generated by the space of Gaussian entropic vectors. It also suggests that studying Gaussian random variables for n ges 4 may be a fruitful approach to studying the space of entropic vectors for arbitrary n.