Starting from concentration of measure hypotheses on $m$ random vectors $Z_1,\ldots, Z_m$, this article provides an expression of the concentration of functionals $\phi(Z_1,\ldots, Z_m)$ where the variations of $\phi$ on each variable depend on the product of the norms (or semi-norms) of the other variables (as if $\phi$ were a product). We illustrate the importance of this result through various generalizations of the Hanson-Wright concentration inequality as well as through a study of the random matrix $XDX^T$ and its resolvent $Q = (I_p - \frac{1}{n}XDX^T)^{-1}$, where $X$ and $D$ are random, which have fundamental interest in statistical machine learning applications.

翻译：本文从m个随机向量Z_1,...,Z_m的测度集中假设出发，给出了泛函φ(Z_1,...,Z_m)的集中性表达式，其中φ在每个变量上的变化依赖于其他变量范数（或半范数）的乘积（即φ表现为乘积形式）。我们通过Hanson-Wright集中不等式的多种推广形式，以及对随机矩阵XDX^T及其预解式Q=(I_p - 1/n XDX^T)^{-1}的研究（其中X和D均为随机变量），阐明了该结果在统计机器学习应用中的基础性重要意义。