Large Deviation Probabilities for Sums of Independent Random Variables

Abstract

The material presented in this chapter is unique to the present text. After an introductory discussion of the concept and importance of large deviation probabilities, Cramér’s condition is introduced and the main properties of the Cramér and Laplace transforms are discussed in Sect. 9.1. A separate subsection is devoted to an in-depth analysis of the key properties of the large deviation rate function, followed by Sect. 9.2 establishing the fundamental relationship between large deviation probabilities for sums of random variables and those for sums of their Cramér transforms, and discussing the probabilistic meaning of the rate function. Then the logarithmic Large Deviations Principle is established. Section 9.3 presents integro-local, integral and local theorems on the exact asymptotic behaviour of the large deviation probabilities in the so-called Cramér range of deviations. Section 9.4 is devoted to analysing various types of the asymptotic behaviours of the large deviation probabilities for deviations at the boundary of the Cramér range that emerge under different assumptions on the distributions of the random summands. In Sect. 9.5, the behaviour of the large deviation probabilities is found in the case of heavy-tailed distributions, namely, when the distributions tails are regularly varying at infinity. These results are used in Sect. 9.6 to find the asymptotics of the large deviation probabilities beyond the Cramér range of deviations, under special assumptions on the distribution tails of the summands.KeywordsLarge Deviation ProbabilitiesLarge Deviation Rate FunctionLocal TheoremsRandom SummandsExact Asymptotic BehaviorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.