Speed and concentration of the covering time for structured coupon collectors

Let $V$ be an $n$-set, and let $X$ be a random variable taking values in the powerset of $V$. Suppose we are given a sequence of random coupons $X_1, X_2, \ldots$, where the $X_i$ are independent random variables with distribution given by $X$. The covering time $T$ is the smallest integer $t\geq 0$ such that $\bigcup_{i=1}^tX_i=V$. The distribution of $T$ is important in many applications in combinatorial probability, and has been extensively studied. However the literature has focussed almost exclusively on the case where $X$ is assumed to be symmetric and/or uniform in some way. In this paper we study the covering time for much more general random variables $X$; we give general criteria for $T$ being sharply concentrated around its mean, precise tools to estimate that mean, as well as examples where $T$ fails to be concentrated and when structural properties in the distribution of $X$ allow for a very different behaviour of $T$ relative to the symmetric/uniform case.