Refined Large Deviation Limit Theorems

Description

This is a developing area of modern probability theory, which has applications in many areas. This volume is devoted to the systematic study of results on large deviations in situations where Cramér's condition on the finiteness of exponential moments may not be satisfied Cover Title Page Copyright Page Dedication Page Table of Contents Foreword Introduction Chapter 1 Asymptotic Expansions Taking into Account the Cases when the Number of Summands Comparable with the Sum is Less than or Equal to Two. 1.1 Upper Estimates for (P{Sn > y] - n.ca1 .ya1 | 1.2 Asymptotic Expansions of the Probabilities of Large Deviations of Sn Taking into Account the Case when Two Summands are Comparable with the Sum 1.3 Asymptotic Expansions of the Probabilities of Large Deviations of Sn in the Case of Quite Asymmetric Constraints on the Asymptotic Behavior of the Tails Chapter 2 Asymptotic Expansions of the Probabilities of Large Deviations and Non-Uniform Estimates of Remainders in CLT. 2.1 The Case of Power Tails with Integer Index a1 > 3 2.2 The Case of Power Tails with Index a1 = 2 Chapter 3 Asymptotic Expansions Taking into Account the Cases when the Number of Summands Comparable with the Sum Does not Exceed a Fixed Integer. 3.1 Recursive Construction of Asymptotic Expansions of P{Sn > y} in the Case of a Non-Normal Stable Law 3.2 Asymptotic Expansions of the Probabilities of Large Deviations of Sn and Non-Uniform Estimates of Remainders in Limit Theorems on Weak Convergence to Non-Normal Stable Laws 3.3 Proof of Proposition 3.1.1 Chapter 4 Limit Theorems on Large Deviations for Order Statistics. 4.1 Large Deviations for Maxima: the Tail Approximation/Extreme Value Approximation Alternative 4.2 Large Deviations for Trimmed Sums 4.3 Conditional Limit Theorems on Large Deviations for Trimmed Sums 4.4 Conditional Limit Theorems on Weak Convergence for Trimmed Sums Chapter 5 Large Deviations for I.I.D. Random Sums when Cramer's Condition is Fulfilled Only on a Finite Interval. 5.1 Exact Asymptotics of the Probabilities of Large Deviations and of the Expectations of Smooth Functions of I.I.D. Random Sums in the Case of Exponential-Power Tails. 5.2 Rough Asymptotics of the Probabilities of Large Deviations for I.I.D.Random Sums in the Case when Cramer's Condition is Fulfilled Only on a Finite Interval. 5.3 Discontinuity of the Most Typical Paths of Random Step-Functions and the Generalized Concept of the Action Functional. 5.4 On Martingale Methods for the Derivation of Upper Estimates of the Probabilities of Large Deviations of I.I.D. Random Sums. References