Unbounded Probability Theory and Its Applications

TL;DR

This paper introduces an extended probability framework for infinitely increasing random variables, connecting it with thermodynamics and exploring applications in economics, networks, and self-learning systems.

Contribution

It develops a novel approach to probability theory for unbounded variables, linking it with thermodynamics and nonstandard analysis, and demonstrates diverse practical applications.

Findings

Established a relationship between mathematical expectation and thermodynamic temperature.

Connected the theory with Van-der-Waals law of corresponding states.

Applied the concepts to economics, internet networks, and self-teaching systems.

Abstract

The paper deals with the order statistics and empirical mathematical expectation (which is also called the estimate of mathematical expectation in the literature) in the case of infinitely increasing random variables. The Kolmogorov concept which he used in the theory of complexity and the relationship with thermodynamics which was pointed out already by Poincar\'e are considered. The mathematical expectation (generalizing the notion of arithmetical mean, which is generally equal to infinity for any increasing sequence of random variables) is compared with the notion of temperature in thermodynamics by using an analog of nonstandard analysis. The relationship with the Van-der-Waals law of corresponding states is shown. Some applications of this concept in economics, in internet information network, and self-teaching systems are considered.