1. komponenta

Lecture type	Total
Lectures	45

* Load is given in academic hour (1 academic hour = 45 minutes)

COURSE AIMS AND OBJECTIVES: The goal of the course is to introduce the basics of mathematical analysis in the set of real numbers.

COURSE DESCRIPTION AND SYLLABUS:
1. Introduction: Sigma algebras and rings. Measure and probability.
2. Random variables and their distributions: Distributions of random variables and vectors. Integrals and expectations of random variables. Independence. Borel-Cantelli lemmas and Kolmogorov's law 0-1.
3. Moment and laws of large numbers: Moments, moment inequalities and convergence of random variables. The weak law of large numbers. The strong law of large numbers.
4. Convergence in distribution and coupling: The concept of coupling. Convergence in distribution and in total variation. Chen-Stein method. Convergence towards the Poisson distribution. Convergence towards the Normal distribution. Extensions and applications of central limit theorems. Cramer-Wold device.
5. Additional topics: Applications of probability models. Introduction to the theory of large deviations.

1. Probability and measure, Billingsley, Patrick, John Wiley & Sons, 2008.
2. Teorija vjerojatnosti, Sarapa, Nikola, Školska knjiga, 2002.
3. Probability: an introduction, Grimmett, Geoffrey, and Dominic Welsh, Oxford University Press, 2024.
4. Probability: theory and examples. Vol. 49, Durrett, Rick, Cambridge university press, 2019.
5. Foundations of modern probability. 3rd edition, Kallenberg, Olav, Springer, 2021.

Mandatory course - Regular study - Financial and Business Mathematics