2 Preliminaries 7 2.1 Matrices and their singular values . . . . . . . . . . . . . . . . . . 7 2.2 Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Sub-gaussian random variables . . . . . . . . . . . . . . . . . . . 9 2.4 Sub-exponential random variables . . . . . . . . . . . . . . . . . . 14 2.5 Isotropic random vectors . . . . . . . . . . . . . . . . . . . . . . ...

1.0 Rayleigh Distribution Using central limit theorem arguments, one can show that the I and Q channels on a mobile radio multipath fading channel are independent Gaussian (normal) random variables. Jakes [1] and others show that the envelope of two independent and identically distributed (iid) Gaussian random variables is Rayleigh distributed.1 Probability Density Function (pdf) (usual form fo...

The “covariance” of complex random variables and processes, when defined consistently with the corresponding notion for real random variables, is shown to be determined by the usual (complex) covariance together with a quantity called the pseudo-covariance. A characterization of uncorrelatedness and wide-sense stationarity in terms of covariance and pseudocovariance is given. Complex random var...

We obtain a representation theorem for Banach space valued Gaussian random variables as integrals against a white noise. As a corollary we obtain necessary and sufficient conditions for the existence of a white noise representation for a Gaussian random field indexed by a measure space. We then show how existing theory for integration with respect to Gaussian processes indexed by [0, 1] can be ...

in this paper, we present the numerical solution of ordinary dierential equations (or sdes), from each order especially second-order with time-varying and gaussian random coecients. we indicate a complete analysis for second-order equations in special case of scalar linear second-order equations (damped harmonic oscillators with additive or multi- plicative noises). making stochastic dierent...

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...