The Double Dixie Cup Problem is similar to the Coupon collector's problem where a collector buys a random card out of $n$ at a time, but its goal is to collect $m$ copies of each card (the Coupon Collector is the special case of $m=1$).
In the above reference, Newman and Shepp bound the expected number of coupons the collector needs to buy before completing the $m$ sets and show that it is $$ n \log n + (m-1) n \log\log n + O(n). $$
I'm looking for a high-probability bound $T(\delta)$ such that the number of required coupons $C_{n,m}$ satisfies: $$ \Pr[C_{n,m} > T(\delta)] \le \delta. $$
Any thoughts on how to derive this?