Limit Theorems for Random Cubical Homology

Abstract

This paper studies random cubical sets in \(\mathbb {R}^d\). Given a cubical set \(X \subset \mathbb {R}^d\), a random variable \(\omega _Q\in [0,1]\) is assigned for each elementary cube Q in X, and a random cubical set X(t) is defined by the sublevel set of X consisting of elementary cubes with \(\omega _Q\le t\) for each \(t\in [0,1]\). Under this setting, the main results of this paper show the limit theorems (law of large numbers and central limit theorem) for Betti numbers and lifetime sums of random cubical sets and filtrations. In addition to the limit theorems, the positivity of the limiting Betti numbers is also shown.