Variance of Longest Run Duration in a Random Bitstring

TL;DR

This paper investigates the uncertainty and correlation properties of longest run durations and bit sums in random bitstrings, extending previous work with a focus on variance and asymptotic behavior using experimental methods.

Contribution

It introduces an analysis of the variance and correlation of longest run lengths and bit sums in various configurations of random bitstrings, emphasizing experimental approaches.

Findings

Negative correlations diminish as bitstring length increases for clumped 1s.

Nonzero limits observed for correlations involving separated 1s.

Experimental methods effectively analyze asymptotic behaviors.

Abstract

We continue an earlier study, starting with unconstrained $n$-bitstrings, focusing now less on average behavior and more on uncertainty. The interplay between $\bullet$ longest runs of 0s and of 1s, when bitstrings are multus $\bullet$ longest runs of 0s and bitsums (# of 1s), when bitstrings are solus $\\$ is examined. While negative correlations approach zero as $n \rightarrow \infty$ in the former (for clumped 1s), the limit is evidently nonzero in the latter (for separated 1s). Similar analysis is possible when both 0s and 1s are clumped (bimultus), and when 0s are clumped but 1s are separated (persolus). Our methods are experimentally-based.