We discuss non-Gaussian random matrices whose elements are random variables with heavy-tailed probability distributions. In probability theory heavy tails of the distributions describe rare but violent events which usually have dominant influence on the statistics. They also completely change universal properties of eigenvalues and eigenvectors of random matrices. We concentrate here on the uni...

We compute explicit bounds in the Gaussian approximation of functionals of infinite Rademacher sequences. Our tools involve Stein’s method, as well as the use of appropriate discrete Malliavin operators. Although our approach does not require the classical use of exchangeable pairs, we employ a chaos expansion in order to construct an explicit exchangeable pair vector for any random variable wh...

In this section, we use our filter technique to give a near sample-optimal computationally efficient algorithm to robustly estimate the mean of a sub-gaussian density with a known covariance matrix, thus proving Theorem 3.1. We emphasize that the algorithm and its analysis is essentially identical to the filtering algorithm given in Section 8.1 of [DKK+16] for the case of a Gaussian N (μ, I). T...

The asymptotic probability density function of nonlinear phase noise, often called the Gordon-Mollenauer effect, is derived analytically when the number of fiber spans is very large. The nonlinear phase noise is the summation of infinitely many independently distributed noncentral chi-square random variables with two degrees of freedom. The mean and standard deviation of those random variables ...

This paper is concerned with the construction of atomic Gaussian multiplicative chaos and the KPZ formula in Liouville quantum gravity. On the first hand, we construct purely atomic random measures corresponding to values of the parameter γ2 beyond the transition phase (i.e. γ2 > 2d) and check the duality relation with sub-critical Gaussian multiplicative chaos. On the other hand, we give a sim...

This paper considers a multi-user multiple-input multiple-output (MU-MIMO) system with independent phase noise (PN) at user terminals (UTs) due to non-synchronous noisy local oscillators. The conventional factor graph of MU-MIMO involves an observation multiple high-dimensional variables in the form multiplication and summation. In this work, is represented by introducing auxiliary variables, w...

In one-bit compressed sensing, previous results state that sparse signals may be robustly recovered when the measurements are taken using Gaussian random vectors. In contrast to standard compressed sensing, these results are not extendable to natural non-Gaussian distributions without further assumptions, as can be demonstrated by simple counter-examples involving extremely sparse signals. We s...