On the number of successes and their series in Bernoulli sequences of random variables

S. M. Ananjevskii ; V. B. Nevzorov

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 34, Tome 525 (2023), pp. 22-29

Résumé

One of the classical directions in probability theory is research related to the so-called Bernoulli sequences of random variables. We are talking about independent random variables $ X_1, X_2,\ldots,$ taking the value $1$ with some probability $p, 0$ and the value $0$ with probability $q=1-p.$ Often the event $\{X_n=1\}$ is interpreted as "success in the $n$th trial", and its complement—the event $\{X_n=0\}$ — as “failure in this trial”. This classic scheme is named after Jacob Bernoulli, who considered such sequences. His results were published by his nephew Nicolaus Bernoulli in 1713 in Basel in the book Ars Conjectandi (“The Art of Conjecture”). Since then, although many results have appeared for various constructions based on Bernoulli random variables, there is a need to consider new schemes for them and to solve new problems. The article examines the relationship of such two-point distributions with a number of other probabilistic laws. A small review of previous results in this area and some new ones are added. The research started in previous works of the authors is continued.