CHEBYSHEV-TYPE INEQUALITIES AND LARGE DEVIATION PRINCIPLES.

Let ξ1, ξ2, . . . be a sequence of independent copies of a random variable (r.v.) ξ, ... is the Legendre transform of A(λ). In this paper, which is partially a review to some extent, we consider generalization of the exponential Chebyshev-type inequalities P(Sn ≥ αn) ≤ exp{-nΛ(α)}, α ≥ Eξ, 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--Réenyi 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. [ABSTRACT FROM AUTHOR]