Selected Topics in Characteristic Functions

Abstract

Part 1 Basic properties: definition and elementary properties continuity theorems and inversion formulas criteria inequalities characteristic functions and movements, expansion of characteristic functions, asymptotic behaviour unimodality analycity of characteristic functions multivariate characteristic functions. Part 2 Inequalities: auxiliary results inequalities for characteristic functions of distributions with bounded support moment inequalities estimates for the characteristic functions of unimodal distributions estimates for the characteristic functions of absolutely continuous distributions estimates for the characteristic functions of discrete distributions inequalities for multivariate characteristic functions inequalities involving integrals of characteristic functions inequalities involving differences of characteristic functions estimates for the positive zero of a characteristic function. Part 3 Empirical characteristic functions: definition and basic properties asymptotic properties of empirical characteristic functions the first positive zero parameter estimation non-parametric density estimation 1 non-parametric density estimation 2 tests for independence tests for symmetry testing for normality goodness-of-fit tests based on empirical characteristic functions.