A probability inequality for obtaining lower bounds in the large deviation principle
The universal tool for obtaining the upper bounds in the large deviation principle is the Chebyshev inequality. The aim of the article is to obtain a probability inequality which might estimate the probabilities \({\mathbf P}(\xi\in A)\) from below in cases when the latter are small, and which could be used for proving lower bounds in the large deviation principle. Such an inequality is proved, and the lower bound is expressed in terms of the deviation function and probabilities \({\mathbf P}(\xi^\lambda\in A)\), where \(\xi^\lambda\) is a random variable conjugate to \(\xi\) (via Cramér transform).
scientific article; zbMATH DE number 1023930
scientific article