Some superconcentration inequalities for extrema of stationary Gaussian Processes

具有独立元素的极值随机矩阵与矩阵超集中不等式

Extremal random matrices with independent entries and matrix superconcentration inequalities

该研究证明了元素独立中心化的（次）高斯随机矩阵谱范数的非渐近集中不等式，改进了已有结果，通过极值问题得出最优尾行为，并得到高斯Wishart矩阵高阶矩的精确界。

Non-asymptotic variance bounds and deviation inequalities by optimal transport

The purpose of this note is to show how simple Optimal Transport arguments, on the real line, can be used in Superconcentration theory. This methodology is efficient to produce sharp non-asymptotic variance bounds for various functionals (maximum, median, $l^p$ norms) of standard Gaussian random vectors in $\R^n$. The flexibility of this approach can also provide exponential deviation inequalities reflecting preceding variance bounds. As a further illustration, usual laws from Extreme theory and Coulomb gases are studied.

Extremals in Hardy-Littlewood-Sobolev inequalities for stable processes

We prove the existence of an extremal function in the Hardy-Littlewood-Sobolev inequality for the energy associated to an stable operator. To this aim we obtain a concentration-compactness principle for stable processes in $\mathbb{R}^N$.

Concentration inequalities for dependent Random variables via the martingale method

The martingale method is used to establish concentration inequalities for a class of dependent random sequences on a countable state space, with the constants in the inequalities expressed in terms of certain mixing coefficients. Along the way, bounds are obtained on martingale differences associated with the random sequences, which may be of independent interest. As applications of the main result, concentration inequalities are also derived for inhomogeneous Markov chains and hidden Markov chains, and an extremal property associated with their martingale difference bounds is established. Thi…