Chebyshev-type inequalities and large deviation principles

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 Erdős–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.