A binomial splitting process in connection with corner parking problems

TL;DR

This paper introduces a binomial splitting process that models complex combinatorial problems like corner parking, digital trees, and urn occupancy, revealing oscillating and non-convergent distribution behaviors.

Contribution

It develops a novel binomial splitting process framework and analyzes its distributional properties, connecting it to various combinatorial and probabilistic models.

Findings

Distribution has logarithmic mean and bounded oscillating variance.

Oscillations occur when binomial parameter p ≠ 1/2.

No limiting distribution exists due to periodic fluctuations.

Abstract

A special type of binomial splitting process is studied. Such a process can be used to model a high-dimensional corner parking problem, as well as the depth of random PATRICIA tries (a special class of digital tree data structures). The later also has natural interpretations in terms of distinct values in iid geometric random variables and the occupancy problem in urn models. The corresponding distribution is marked by logarithmic mean and bounded variance, which is oscillating, if the binomial parameter $p$ is not equal to 1/2, and asymptotic to 1 in the unbiased case. Also, the limiting distribution does not exist owing to periodic fluctuations.