Series of Successes in Bernoulli Sequences of Random Variables

Some of the first studies in the probability theory were related to Bernoulli schemes, viz. sequences of independent identically distributed random variables X1, X2 taking the values 1 with some probability 0 < p < 1 and 0 with the probability q = 1 – p. Frequently, for the sake of convenience, the events {Xk = 0} and {Xk = 1}, k = 1, 2, … were interpreted as “failure” and “success” in the kth test. The sums Sn = X1 + X2 + … + Xn specified the number of successes in n tests and had the binomial B(n, p)-distribution. Studying the sequences of such random variables led to the need to deal with geometrically distributed random variables. Limit theorems for properly centered and normalized sums required one to consider and study normally distributed random variables. Working with classical Bernoulli sequences and their other two-point generalizations led to the need to develop various methods to study them, which were then used for other random variables as well. Despite numerous results obtained since the publication of Jacob Bernoulli’s The Art of Conjecturing (Ars Conjectandi) in 1713 [1], new schemes for Bernoulli variables keep appearing and require further study. This work continues to explore various generalizations of the Bernoulli scheme. Some new problems are considered that are associated with various series of successes in Bernoulli sequences. This work is a continuation of the authors' articles published in 2022, 2023, and 2024.