Characteristic functions (CFs) are often used in problems involving convergence in distribution, independence of random variables, infinitely divisible distributions, and stochastics. The most famous use of characteristic functions is in the proof of the Central Limit Theorem, also known as the Fundamental Theorem of Statistics. Though less frequent, CFs have also been used in problems of nonparametric time series analysis and in machine learning. Moreover, CFs uniquely determine their distribution, much like the moment generating functions (MGFs), but the major difference is that CFs always exists, whereas MGFs can fail, e.g. the Cauchy distribution. This makes CFs more robust in general.
In the following, I will present an introduction and basic properties of the Fourier-Stieltjes transform, its inverse and relation to the Radon-Nikodym derivative, then go on to prove the Lévy Continuity Theorem, and finally a short presentation of measure convolutions. Much of the following presentation will be for probability measures and their distribution functions; however, some results can be generalized to (un)signed finite measures. One can find an overview of background knowledge in the Appendix.
Vandegriffe, Austin G., "A Brief on Characteristic Functions" (2020). Graduate Student Research & Creative Works. 2.
Mathematics and Statistics
Keywords and Phrases
Probability Theory; Harmonic Analysis; Characteristic Functions; Fourier–Stieltjes Transform; Radon Measures
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