Unbounded Probability Theory and Its Applications

Abstract

The order statistics and empirical mathematical expectation (also called the estimate of mathematical expectation in the literature) are considered 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 Poincaré, are considered. We compare the mathematical expectation (which is a generalization of the notion of arithmetical mean, and is generally equal to infinity for any increasing sequence of random variables) with the notion of temperature in thermodynamics while deploying a certain analogue of nonstandard analysis. It is shown that there is a relationship with the Van der Waals law of corresponding states. A number of applications of this concept in economics, in internet information networks, and self-teaching systems are also considered.