Unbounded probability theory and its applications (Q2863589)

The paper under review is a continuation of studies conducted in [\textit{V. P. Maslov}, Theory Probab. Appl. 48, No. 2, 359--367 (2003); translation from Teor. Veroyatn. Primen. 48, No. 2, 403--411 (2003; Zbl 1099.91018)] and [Theory Probab. Appl. 48, No. 4, 723--733 (2003); translation from Teor. Veroyatn. Primen. 48, No. 4, 800--810 (2003; Zbl 1089.91026)]. The authors consider the order statistics and the so-called ``empirical mathematical expectation'' 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 by Poincaré. The mathematical expectation is compared with the notion of temperature in thermodynamics while deploying an analogue of nonstandard analysis. It is shown that there is a relationship with the Van der Waals law of corresponding states. The depicted concept is illustrated by its application in economics, in Internet information networks, and in self-teaching systems.