# Mini-course: Regularly varying functions with applications in probability theory

Professor O.I. Klesov from National Technical University of Ukraine "Igor Sikorsky Kyiv Politechnic Institute" will give a mini-course on the following topic:

We discuss the notion of regularly varying functions and some applications in probability theory. Some of the topics to be discussed are in order. Note however that not all topics will be discussed in full detail. The final choice of topics will depend on the time available.

PART 1: \(\mathit{RV}\) FUNCTIONS IN LIMIT THEOREMS

- Central limit theorem
- stable limit laws
- Relative stability (Saint-Petersburg paradox)
- Extremes
- Records

PART 2: \(\mathit{RV}\) FUNCTIONS AND CAUCHY FUNCTIONAL EQUATION

- Additive functions
- Differentiable additive functions
- Continuous additive functions
- Monotone additive functions
- Non-measurable additive functions

PART 3: FURTHER APPLICATIONS OF \(\mathit{RV}\) FUNCTIONS

- Interest rate
- Memory less distributions
- Distribution of attaining zero for stochastic processes
- Buffon neddle problem
- Formal justice

PART 4: BASIC RESULTS FOR \(\mathit{RV}\) FUNCTIONS

- Integral representation
- Uniform convergence theorem
- Characterization theorem