The central limit theorem ensures that a sum of random variables tends to a Gaussian distribution as their total number tends to infinity. However, for a class of positive random variables, we find that the sum tends faster to a log-normal distribution. Although the sum tends eventually to a Gaussian distribution, the distribution of the sum is always close to a log-normal distribution rather t...

1. Abstract For the performance measure approach (PMA) of RBDO, a transformation between the input random variables and the standard normal random variables is required to carry out the inverse reliability analysis. Since the transformation uses the joint cumulative density function (CDF) of input variables, the joint CDF should be known before carrying out RBDO. In many industrial RBDO problem...

We de ne a covariance-type operator on Wiener space: for F and G two random variables in the Gross-Sobolev space D of random variables with a square-integrable Malliavin derivative, we let ΓF,G:= ⟨ DF,−DL−1G ⟩ , where D is the Malliavin derivative operator and L−1 is the pseudo-inverse of the generator of the Ornstein-Uhlenbeck semigroup. We use Γ to extend the notion of covariance and canonica...

We define a covariance-type operator on Wiener space: for F and G two random variables in the Gross-Sobolev space D of random variables with a square-integrable Malliavin derivative, we let ΓF,G:= 〈 DF,−DL−1G 〉 where D is the Malliavin derivative operator and L−1 is the pseudo-inverse of the generator of the Ornstein-Uhlenbeck semigroup. We use Γ to extend the notion of covariance and canonical...

This article derives the probability density function (pdf) of the sum of a normal random variable and a (sphered) Student’s-t distribution on odd degrees of freedom greater than or equal to three. Apart from its intrinsic interest applications of this result include Bayesian wavelet shrinkage, Bayesian posterior density derivations, calculations in the theoretical analysis of projection indice...

• Proof: X is j G implies that V = uX is G with mean uμ and variance uΣu. Thus its characteristic function, CV (t) = e ituμe−t 2uTΣu/2. But CV (t) = E[e itV ] = E[e TX ]. If we set t = 1, then this is E[e TX ] which is equal to CX(u). Thus, CX(u) = CV (1) = e iuμe−u TΣu/2. • Proof (other side): we are given that the charac function ofX, CX(u) = E[eiuTX ] = e μe−u TΣu/2. Consider V = uX. Thus, C...

We establish that Gaussian distributions are the optimizers for a particular optimization problem related to determining the hypercontractivity parameters for a pair of jointly Gaussian random variables.

Simulations are often needed when the performance of new methods is evaluated. If the method is designed to be blind or robust, simulation studies must cover the whole range of potential random input. It follows that there is a need for advanced tools of data generation. The purpose of this paper is to introduce a technique for the generation of correlated multivariate random data with non-Gaus...

The complete convergence is investigated for moving-average processes of doubly infinite sequence of negative dependence sub-gaussian random variables with zero means, finite variances and absolutely summable coefficients. As a corollary, the rate of complete convergence is obtained under some suitable conditions on the coefficients.

The particle swarm optimization (PSO), now widely used in the electromagnetics community, is a robust evolutionary method based on the property of swarm intelligence. This paper focuses on the random variable effect in the PSO algorithm, and two random distribution functions, namely, the uniform distribution and Gaussian distribution, are studied and compared in details. It is revealed through ...