What can we say about the complexity of generating (negative) binomial and (negative) hypergeometric random variates? In particular, it is possible to generate (negative) binomial and (negative) hypergeometric variates in (expected) constant time (i.e. independent of the distribution parameters)?
There is quite a bunch of literature; however, it’s hard to understand a lot of the papers. Moreover, I found some statements that seem contradictory to me (probably due to wrong understanding).
For example, Stadlober (or similar here) mentions a "generalization [of the ratio-of-uniforms approach] to any unimodal discrete distribution". The ratio-of-uniforms approach has been called uniformly fast, which is a synonym for constant time, I suppose (?). That would mean that we can generate random variates for each discrete distribution in (expected) constant time.
However, in another paper, I found the following theorem: On a RAM with word size $w$, any algorithm sampling a geometric random variate $Geo(p)$ with parameter $p \in (0,1)$ needs at least expected runtime $\Omega(1 + \log(1/p)/w)$.
Apparently, this means that it is not possible to generate negative binomial variates in time independent of the distribution parameters.