i=1 ǫiAi (1) of deterministic Hermitian matrices A1, . . . , An multiplied by random coefficients. Recall that a Rademacher sequence is a sequence {ǫi}i=1 of i.i.d. random variables with ǫ1 uniform over {−1,+1}. A standard Gaussian sequence is a sequence i.i.d. standard Gaussian random variables. Our main goal is to prove the following result. Theorem 1 (proven in Section 3) Given positive inte...

1 Definition The topic in this lecture is Subgaussian random variables. We start with the definition, and discuss some properties they hold. Definition 1 (Subgaussian random variables). A random variable X is subgaussian if ∃c, C such that P(|x| > t) ≤ Ce −ct 2 ∀t ≥ 0. (1) As the name suggests, the notion of subgaussian random variables is a generalization of Gaussian random variables. Both the...

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.