Martin–Löf random generalized Poisson processes

Abstract

Martin–Löf randomness was originally defined and studied in the context of the Cantor space 2ω. In [2] probability theoretic random closed sets (RACS) are used as the foundation for the study of Martin–Löf randomness in spaces of closed sets. We use that framework to explore Martin–Löf randomness for the space of closed subsets of R and a particular family of measures on this space, the generalized Poisson processes. This gives a novel class of Martin–Löf random closed subsets of R. We describe some of the properties of these Martin–Löf random closed sets; one result establishes that a real number is Martin–Löf random if and only if it is contained in some Martin–Löf random closed set.