Zapraszamy na wykład w ramach cyklu Hot Topics, który wygłosi prof. Tomasz Kania (Universitet Jagielloński i Czeska Akademia Nauk)
Szanowni Państwo,
mam przyjemność zapowiedzieć kolejny wykład w ramach cyklu Hot Topics, który wygłosi prof. Tomasz Kania (Universitet Jagielloński i Czeska Akademia Nauk). Wykład odbędzie się w trybie zdalnym w dniu 3 lutego o godz. 14:15. Tytuł wykładu:
A no-limits approach to Probability and Statistics: an Introduction to Non-Standard Analysis and its applications to risk estimation
Abstract. Non-standard analysis (NSA) offers a strikingly different way of thinking about calculus, probability, and statistics: instead of taking limits, one works directly with infinitesimal and infinite quantities. In this talk, I will give a gentle but concrete introduction to NSA and explain how it can be used as a practical tool rather than a purely foundational curiosity. We begin with the ultrapower construction of the hyperreal numbers and the transfer principle, illustrating the familiar slogan that in NSA one can often do analysis without limits. Classical notions such as derivatives, integrals, laws of large numbers, and central limit theorems can be reformulated using infinite sums over hyperfinite sets and then recovered by taking standard parts. Along the way, we discuss the Loeb measure construction, which turns hyperfinite probability spaces into genuine, σ-additive probability spaces and provides a powerful bridge between discrete and continuous probability. In the second part of the talk, we move from foundations to applications. We introduce two standard risk measures used in finance and insurance; Value at Risk (VaR) and Expected Shortfall (ES), and explain their interpretation from both probabilistic and statistical viewpoints. Using the hyperfinite perspective, empirical distributions and order statistics arise naturally as shadows of infinite hyperfinite samples. This leads to clean, intuitive constructions of risk estimators and transparent proofs of consistency, asymptotic normality, and bootstrap validity. The overall message of the talk is that non-standard analysis provides a unifying “probability-to-statistics dictionary”: population-level quantities are evaluated on infinite hyperfinite samples, and finite-sample estimators appear as finite shadows of this picture. No prior knowledge of model theory or non-standard analysis is assumed; the emphasis is on intuition, examples, and conceptual clarity rather than technical detail.
Adres strony internetowej prof. Kani: https://users.math.cas.cz/~kania/#home
Pozdrawiam serdecznie,
Jacek Jachymski