Chebyshev-type inequalities and large deviation principles

A. A. Borovkov ; A. V. Logachov ; A. A. Mogul'skii

Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 4, pp. 718-733

Résumé

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.

Détail

Keywords: exponential Chebyshev-type inequality, large deviation principle, local large deviation principle, random walk, random field, Erdős–Rényi graphs.