Unbounded Probability Theory and Its Applications

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.