  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.
