Theory of Probability &Its Applications, Volume 66, Issue 4, Page 570-581, February 2022.
Let $\xi_1,\xi_2,\dots$ be a sequence of independent copies of a random variable (r.v.) $\xi$, ${S_n=\sum_{j=1}^n\xi_j}$, $A(\lambda)=\ln\mathbf{E}e^{\lambda\xi}$, $\Lambda(\alpha)=\sup_\lambda(\alpha\lambda-A(\lambda))$ is the Legendre transform of $A(\lambda)$. In this paper, which is partially a review to some extent, we consider generalization of the exponential Chebyshev-type inequalities $\mathbf{P}(S_n\geq\alpha n)\leq\exp\{-n\Lambda(\alpha)\}$, $\alpha\geq\mathbf{E}\xi$, for the following three cases: I. Sums of random vectors, II. stochastic processes (the trajectories of random walks), and III. random fields associated with Erd\Hos--Rényi graphs with weights. It is shown that these generalized Chebyshev-type inequalities enable one to get exponentially unimprovable upper bounds for the probabilities to hit convex sets and also to prove the large deviation principles for objects mentioned in I--III.

中文翻译：

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Chebyshev 型不等式和大偏差原理

概率论及其应用，第 66 卷，第 4 期，第 570-581 页，2022 年 2 月。
令 $\xi_1,\xi_2,\dots$ 为随机变量 (rv) 的独立副本序列 $\xi$, ${S_n=\sum_{j=1}^n\xi_j}$, $A( \lambda)=\ln\mathbf{E}e^{\lambda\xi}$, $\Lambda(\alpha)=\sup_\lambda(\alpha\lambda-A(\lambda))$ 是勒让德变换$A(\lambda)$。在本文中，在某种程度上是部分回顾，我们考虑指数 Chebyshev 型不等式 $\mathbf{P}(S_n\geq\alpha n)\leq\exp\{-n\Lambda(\alpha )\}$, $\alpha\geq\mathbf{E}\xi$，对于以下三种情况： I. 随机向量的和，II. 随机过程（随机游走的轨迹），以及 III。与 Erd\Hos--Rényi 图相关的随机字段，带有权重。