Covering a compact space by fixed-radius or growing random balls

Abstract

Simple random coverage models, well studied in Euclidean space, can also be defined on a general compact metric space. By analogy with the geometric models, and with the discrete coupon collector's problem and with cover times for finite Markov chains, one expects a weak concentration bound for the distribution of the cover time to hold under minimal assumptions. We give two such results, one for random fixed-radius balls and the other for sequentially arriving randomly-centered and deterministically growing balls. Each is in fact a simple application of a different more general bound, the former concerning coverage by i.i.d. random sets with arbitrary distribution, and the latter concerning hitting times for Markov chains with a strong monotonicity property. The growth model seems generally more tractable, and we record some basic results and open problems for that model.