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

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