The Euler Constant as an Entropic Characteristic of Random Binary Sequences

This paper considers an approach to the analysis of random binary sequences based on the introduction of composite events, defined as monotonic runs of elementary events. It is shown that the transition from the analysis of individual bits to the analysis of composite events leads to the emergence of new structural properties of probability that are not described by classical probability theory. Within the framework of the Combinatorics of Long Sequences (CLS), entropy measures for composite and polar composite events are introduced, for which a stable relationship with the Euler constant is established. The obtained results may be used in the analysis of structural properties of random processes, including binary streams of physical origin.