1. By using the sampling theorem to expand N (t − τ, ω) and h(τ) with respect to τ we obtain: Z(t, ω) = τ k∈Z h(kT) sinc π τ − kT T k∈Z N (t − kT, ω) sinc π τ − kT T dτ = T k∈Z h(kT)N (t − kT, ω) where we used orthogonality of the set of functions {sinc (π(τ − kT)/T)} k∈Z. 2. Since N (τ) is a complex Gaussian random process, for each finite set of epochs t 1 , ..., t K , the random variables {N...

Because of random nature of many dependent variables in coastal engineering, treatment of effective parameters is generally associated with uncertainty. Numerical models are often used for dynamic analysis of complex structures, including mechanical systems. Furthermore, deterministic models are not sufficient for exact anticipation of structure’s dynamic response, but probabilistic models...

In this paper we combine the appealing properties of the stable Paretian distribution to model the heavy tails and the GARCH model to capture the phenomenon of the volatility clustering. We assume the asset-returns to have a particular multivariate stable distribution, i.e., to be sub-Gaussian random vectors. In this way the characteristic function has a tractable expression and the density fun...

1. The multivariate normal distribution Let X := (X1 � � � � �X�) be a random vector. We say that X is a Gaussian random vector if we can write X = μ + AZ� where μ ∈ R, A is an � × � matrix and Z := (Z1 � � � � �Z�) is a �-vector of i.i.d. standard normal random variables. Proposition 1. Let X be a Gaussian random vector, as above. Then, EX = μ� Var(X) := Σ = AA� and MX(�) = e � μ+ 1 2 �A���2 =...

We show that the gain for using a waterfilling power allocation instead of a flat allocation over non-singular channel components is negligible at high signal-to-noise ratios. Consider the standard additive noise communication channel with a quadratic power constraint. Although waterfilling provides the optimal input distribution for Gaussian noise channels, sub-optimal distributions are often ...

Known Bernstein-type upper bounds on the tail probabilities for sums of independent zero-mean sub-exponential random variables are improved in several ways at once. The new have a certain optimality property.

We consider the problem of measurement of optical transverse profile parameters and their conjugate variable. Using multimode analysis, we introduce the concept of detection noise modes. For Gaussian beams, displacement and tilt are a pair of transverse-profile conjugate variables. We experimentally demonstrate the optimal encoding and detection of these variables with a spatial homodyning sche...

We consider the homogenization of parabolic equations with large spatiallydependent potentials modeled as Gaussian random fields. We derive the homogenized equations in the limit of vanishing correlation length of the random potential. We characterize the leading effect in the random fluctuations and show that their spatial moments converge in law to Gaussian random variables. Both results hold...

Consider a Gaussian process {Xt}t∈T for some index set T . This is a collection of jointly Gaussian random variables, meaning that every finite linear combination of the variables has a Gaussian distribution. We will additionally assume that the process is centered, i.e. E(Xt) = 0 for all t ∈ T . It is well-known that such a process is completely characterized by the covariances {E(XsXt)}s,t∈T ...