Spacing Distribution of a Bernoulli Sampled Sequence

TL;DR

This paper analyzes the spacing distribution of a Bernoulli sampled sequence, deriving a closed-form probability mass function and demonstrating convergence to a geometric distribution.

Contribution

It provides a closed-form expression for the spacing distribution and proves convergence to a geometric distribution after Bernoulli sampling.

Findings

Derived the closed-form probability mass function of spacings.

Proved convergence of spacings to a geometric distribution.

Enhanced understanding of spacing behavior in sampled sequences.

Abstract

We investigate the spacing distribution of sequence \[S_n=\left\{0,\frac{1}{n},\frac{2}{n},\dots,\frac{n-1}{n},1\right\}\] after Bernoulli sampling. We describe the closed form expression of the probability mass function of the spacings, and show that the spacings converge in distribution to a geometric random variable.