On Bernoulli trials with unequal harmonic success probabilities

TL;DR

This paper investigates a Bernoulli process with unequal harmonic success probabilities, analyzing success counts, timing between successes, and related distributions, with implications for reliability, sampling, and random walks.

Contribution

It introduces a novel Bernoulli scheme with harmonic success probabilities and explores its properties, asymptotics, and connections to known distributions.

Findings

Asymptotic behavior of success counts and timings

Parameter estimation methods for the model

Connections to Sibuya and Yule-Simon distributions

Abstract

A Bernoulli scheme with unequal harmonic success probabilities is investigated, together with some of its natural extensions. The study includes the number of successes over some time window, the times to (between) successive successes and the time to the first success. Large sample asymptotics, statistical parameter estimation, and relations to Sibuya distributions and Yule-Simon distributions are discussed. This toy model is relevant in several applications including reliability, species sampling problems, record values breaking and random walks with disasters.