It is known that a binomial distribution can be approximated by a normal variable with the same mean and variance, for sufficiently large $n$.
What approximations are known for the negative binomial distribution (the number of successes before the $r$-th failure in draws with replacements) and for the negative hypergeometric distribution (the same without replacements), for sufficiently large $r$?
Note: I am mainly interested in assymetric scenarios, in which the total number of successes is much smaller than the total number of failures.