A cut-invariant law of large numbers for random heaps

TL;DR

This paper establishes a law of large numbers for heap monoids with Bernoulli measures, introducing asynchronous stopping times and cut-invariance, and extends results to sub-additive ergodic theorems.

Contribution

It introduces asynchronous stopping times and cut-invariance in heap monoids, proving a new strong law of large numbers and extending to sub-additive ergodic results.

Findings

Proved a strong law of large numbers for heap monoids with Bernoulli measures.

Introduced asynchronous stopping times and cut-invariance concepts.

Extended results to sub-additive ergodic theorems using Kingman's theorem.

Abstract

Heap monoids equipped with Bernoulli measures are a model of probabilistic asynchronous systems. We introduce in this framework the notion of asynchronous stopping time, which is analogous to the notion of stopping time for classical probabilistic processes. A Strong Bernoulli property is proved. A notion of cut-invariance is formulated for convergent ergodic means. Then a version of the Strong law of large numbers is proved for heap monoids with Bernoulli measures. Finally, we study a sub-additive version of the Law of large numbers in this framework based on Kingman sub-additive Ergodic Theorem.