Runs in Random Sequences over Ordered Sets

TL;DR

This paper derives formulas for the distribution, expected value, variance, and probability generating functions of run lengths in random sequences over both totally and partially ordered sets, including complex cases with mixed distributions.

Contribution

It introduces novel formulas and methods for analyzing run lengths in ordered set sequences, especially for cases with mixed discrete and continuous distributions.

Findings

Derived formulas for expected run length and variance.

Provided a method to compute the PGF for countably series-parallel partial orders.

Proved a strong law of large numbers for run lengths in infinite sequences.

Abstract

We determine the distributions of lengths of runs in random sequences of elements from a totally ordered set (total order) or partially ordered set (partial order). In particular, we produce novel formulae for the expected value, variance, and probability generating function (PGF) of such lengths in the case of an arbitrary total order. Our focus is on the case of distributions with both atoms and diffuse (absolutely or singularly continuous) mass which has not been addressed in this generality before. We also provide a method of calculating the PGF of run lengths for countably series-parallel partial orders. Additionally, we prove a strong law of large numbers for the distribution of run lengths in a particular realization of an infinite sequence.