I have gone through a lot of variation of Coupon Collector's problems, but could not determine which one matches my problem.
Suppose, I have $n$ coupons with distinct probabilities: $(p_1,p_2,\ldots,p_n)$. I want to collect any $d$ distinct coupons from these $n$ coupons (with replacement). What is the expected or maximum number of trials I need? If I can get some upper bound, that is also fine. Can I say that the number of trials required is $O(n)$ to get $x$ distinct coupons with high probability?