Selected Topics in Characteristic Functions

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.