On Series of Successes in Bernoulli Sequences of Random Variables

Some of the first studies in probability theory were associated with Bernoulli schemes, viz. sequences of independent identically distributed random variables taking values of 1 with some probability 0 < p < 1 and 0 with the probability q = 1 – p. Studying sequences of such random variables led to the need to deal with a geometrically distributed random variable. Limit theorems for properly centered and normalized sums required normally distributed random variables to be taken into account and studied. Working with classical Bernoulli sequences and their other two-point generalizations led to the need to develop various methods for studying them, which were then used for other random variables. Despite the numerous results obtained since the publication of Jacob Bernoulli’s The Art of Conjecturing in 1713, new schemes for Bernoulli variables keep appearing and require further study. This work is a direct continuation of our works published in 2022 and 2023.