Strong law of large numbers on graphs and groups

TL;DR

This paper generalizes the strong law of large numbers to graph- and group-valued random elements, introduces methods for computing mean-sets, and establishes related probabilistic inequalities and bounds.

Contribution

It extends classical probabilistic laws to graphs and groups, providing theoretical results and practical algorithms for mean-set computation.

Findings

Generalization of the strong law of large numbers to graphs and groups

Establishment of Chebyshev's inequality analogue for group-valued variables

Development of algorithms for computing mean-sets in graphs

Abstract

We consider (graph-)group-valued random element $\xi$, discuss the properties of a mean-set $\ME(\xi)$, and prove the generalization of the strong law of large numbers for graphs and groups. Furthermore, we prove an analogue of the classical Chebyshev's inequality for $\xi$ and Chernoff-like asymptotic bounds. In addition, we prove several results about configurations of mean-sets in graphs and discuss computational problems together with methods of computing mean-sets in practice and propose an algorithm for such computation.