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...

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...

A symmetric random variable is called a Gaussian mixture if it has the same distribution as the product of two independent random variables, one being positive and the other a standard Gaussian random variable. Examples of Gaussian mixtures include random variables with densities proportional to e−|t| p and symmetric p-stable random variables, where p ∈ (0, 2]. We obtain various sharp moment an...

Abstract In this manuscript we give an extension of the classic Salem–Zygmund inequality for locally sub-Gaussian random variables. As application, concentration roots a Kac polynomial is studied, which main contribution manuscript. More precisely, assume existence moment generating function iid coefficients and prove that there exists annulus width $$\begin{aligned} \text {O}( n^{-2}(\log n)^{...

Now we come to one of the most important and inspiring realizations of freeness. Up to now we have realized free random variables by generators of the free group factors or by creation and annihilation operators on full Fock spaces. This was not very surprising because our definition of freeness was modelled according to such situations. But there are objects from a quite different mathematical...