Quantum Mechanical Approach to the Khintchine and Bochner Criteria for Characteristic Functions

While it is generally accepted that quantum mechanics is a probability theory, its methods differ radically from standard probability theory. We use the methods of quantum mechanics to understand some fundamental aspects of standard probability theory. We show that wave functions and operators do appear in standard probability theory. We do so by generalizing the Khintchine and Bochner criteria for a complex function to be a characteristic function. We show that quantum mechanics clarifies these criteria and suggests generalizations of them.