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It is known that the hypergeometric distribution can be approximated by the binomial distribution (it is also shown here Proof that the hypergeometric distribution with large $N$ approaches the binomial distribution.)

I instead would like to show under which limits one can approximate the negative hypergeometric distribution with the negative binomial distribution. In particular, following the wikipedia notation for the negative hypergeometric for a population with $N$ elements, $K$ of them are successes, then we draw without replacement until we see $r$ failures. The number of drawn successes is $k$ which is distributed with the negative hypergeometric distribution:

$$ \binom{k+r-1}{k}\binom{N-r-k}{K-k}\Big / \binom{N}{K} $$

Now I would like to know what limiting procedure can bring from the negative hypergeometric to the negative binomial. Should one expand all the factorials and then collect likewise terms to make it evident?

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Let $N\to\infty$ and $K\to\infty$ in such a way that $\frac{K}{N}\to p\in(0,\,1)$. Then $\frac{N-K}{N}\to 1-p.$

Show that the pmf of negative hypergeometric distribution converges to pmf of negative binomial: $$\frac{\binom{k+r-1}{k}\binom{N-r-k}{K-k}}{\binom{N}{K}} =\binom{k+r-1}{k} \cdot \frac{\binom{N-r-k}{K-k}}{\binom{N}{K}} \to \binom{k+r-1}{k} p^k(1-p)^r. $$ Expand the fraction: $$ \frac{\binom{N-r-k}{K-k}}{\binom{N}{K}} =\underbrace{\frac{K(K-1)\ldots(K-k+1)}{N(N-1)\ldots(N-k+1)}}_k\,\underbrace{\frac{(N-K)(N-K-1)\ldots(N-K-r+1)}{(N-k)(N-k-1)\ldots(N-k-r+1)}}_r $$ The first multiplier converges to $p^k$, and the second one to $(1-p)^r$.

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