Gaussian Random Matrix Models¶for q -deformed Gaussian Variables

Autoregressive Gaussian Random Vectors of First Order

Chapter 1: Sub-Gaussian Random Variables

where μ = IE(X) ∈ IR and σ = var(X) > 0 are the mean and variance of X . We write X ∼ N (μ, σ). Note that X = σZ + μ for Z ∼ N (0, 1) (called standard Gaussian) and where the equality holds in distribution. Clearly, this distribution has unbounded support but it is well known that it has almost bounded support in the following sense: IP(|X −μ| ≤ 3σ) ≃ 0.997. This is due to the fast decay of the...

On the structure of Gaussian random variables

We study when a given Gaussian random variable on a given probability space (Ω,F , P ) is equal almost surely to β1 where β is a Brownian motion defined on the same (or possibly extended) probability space. As a consequence of this result, we prove that the distribution of a random variable in a finite sum of Wiener chaoses (satisfying in addition a certain property) cannot be normal. This resu...

On a quaternion valued Gaussian random variables

In the present note we show that Polya’s type characterization theorem of Gaussian distributions does not hold. This happens because in the linear form, constituted by the independent copies of quaternion random variables, a part of the quaternion coefficients is written on the right hand side and another part on the left side. This gives a negative answer to the question posed in [1]. Mathemat...

Matrix measures, random moments and Gaussian ensembles

We consider the moment space Mn corresponding to p × p real or complex matrix measures defined on the interval [0, 1]. The asymptotic properties of the first k components of a uniformly distributed vector (S1,n, . . . , Sn,n) ∗ ∼ U(Mn) are studied if n → ∞. In particular, it is shown that an appropriately centered and standardized version of the vector (S1,n, . . . , Sk,n) ∗ converges weakly to...