In this paper, we prove multilevel concentration inequalities for bounded functionals $$f = f(X_1, \ldots , X_n)$$ f = f ( X 1 , … , X n ) of random variables $$X_1, \ldots , X_n$$ X 1 , … , X n that are either independent or satisfy certain logarithmic Sobolev inequalities. The constants in the tail estimates depend on the operator norms of k -tensors of higher order differences of f . We provide applications for both dependent and independent random variables. This includes deviation inequalities for empirical processes $$f(X) = \sup _{g \in {\mathcal {F}}} {|g(X)|}$$ f ( X ) = sup g ∈ F | g ( X ) | and suprema of homogeneous chaos in bounded random variables in the Banach space case $$f(X) = \sup _{t} {\Vert \sum _{i_1 \ne \ldots \ne i_d} t_{i_1 \ldots i_d} X_{i_1} \cdots X_{i_d}\Vert }_{{\mathcal {B}}}$$ f ( X ) = sup t ‖ ∑ i 1 ≠ … ≠ i d t i 1 … i d X i 1 ⋯ X i d ‖ B . The latter application is comparable to earlier results of Boucheron, Bousquet, Lugosi, and Massart and provides the upper tail bounds of Talagrand. In the case of Rademacher random variables, we give an interpretation of the results in terms of quantities familiar in Boolean analysis. Further applications are concentration inequalities for U -statistics with bounded kernels h and for the number of triangles in an exponential random graph model.

中文翻译：

---

通过对数 Sobolev 型不等式的有界泛函的浓度不等式

在本文中，我们证明了随机变量 $$X_1, \ldots , X_n 的有界泛函 $$f = f(X_1, \ldots , X_n)$$ f = f ( X 1 , … , X n ) 的多级浓度不等式$$ X 1 , ... , X n 要么是独立的，要么满足某些对数 Sobolev 不等式。尾部估计中的常数取决于 f 的高阶差分的 k 张量的算子范数。我们提供相关和独立随机变量的应用程序。这包括经验过程的偏差不等式 $$f(X) = \sup _{g \in {\mathcal {F}}} {|g(X)|}$$ f ( X ) = sup g ∈ F | 克 ( X ) | 和 Banach 空间情况下有界随机变量的齐次混沌至上 $$f(X) = \sup _{t} {\Vert \sum _{i_1 \ne \ldots \ne i_d} t_{i_1 \ldots i_d} X_{i_1} \cdots X_{i_d}\Vert }_{{\mathcal {B}}}$$ f ( X ) = sup ‖ ∑ i 1 ≠ … ≠ idti 1 … id X i 1 ⋯ X id ‖乙。后一个应用程序与 Boucheron、Bousquet、Lugosi 和 Massart 的早期结果相当，并提供了 Talagrand 的上尾边界。在 Rademacher 随机变量的情况下，我们根据布尔分析中熟悉的数量来解释结果。