TITLE: How exactly are the beta and gamma distributions related?
QUESTION [16 upvotes]: According to Wikipedia, the Beta distribution is related to the gamma distribution by the following relation:
$$\lim_{n\to\infty}n B(k, n) = \Gamma(k, 1)$$
Can you point me to a derivation of this fact? Can it be generalized? For example, is there a similar relation that results in something other than a constant 1 for the Gamma second parameter? What if we have
$$\lim_{n\to\infty,m\to\infty,n=mb}n B(k, m) $$
That is, the two variables go to infinity while maintaining a constant ratio b.
The reason I'm asking is because I'm trying to figure out how to simplify a hieraerchical bayesian model involving the beta distribution.
(This is my first post; sorry for the math notation, the MathJaX syntax was too daunting, but I'll try to learn)

REPLY [5 votes]: There is another way to view the relationship between the gamma distribution and the beta distribution through the Dirichlet distribution. This post (https://math.stackexchange.com/q/190695) talks about exactly how they are related without the Dirichlet distribution, but here is a slightly broader view:
Let $Z_1, Z_2, \ldots, Z_n$ be independent random variables such that
$$ Z_i \sim \text{Gamma}(\alpha_i,1), \quad i=1,\ldots, n$$
where $\alpha_i \geq 0$ are the shape parameters. If $Y_j$ is defined as follows,
$$Y_j = Z_j/\sum_{i=1}^n Z_i, \quad j=1, \ldots, n$$
then $(Y_1, \ldots, Y_n)$ is Dirichlet distributed with parameter $(\alpha_1, \ldots, \alpha_n)$. When $n=2$, the Dirichlet distribution reduces to the Beta distribution, denoted by $\text{Beta}(\alpha_1, \alpha_2)$. 
Here is a link to Ferguson's paper that mentions the above relationship:
http://www.cis.upenn.edu/~taskar/courses/cis700-sp08/papers/ferguson.pdf