# What happens when one of the concentration parameters of a Dirichlet distribution tends to zero?

If I have a dirichlet distribution with parameter $$\alpha \in \mathbb{R}^k = [\alpha_1, \alpha_2, \cdots, \alpha_k]$$, and then I set the component $$\alpha_k$$ to $$\epsilon$$. As I decrease $$\epsilon$$, will this pdf approximate the pdf of a dirichlet distribution with parameters $$[\alpha_1, \alpha_2, \cdots, \alpha_{k-1}]$$? That is, ignoring the last dimension of the samples from $$\text{Dir}([\alpha_1, \alpha_2, \cdots, \alpha_{k-1}, \epsilon])$$, will they follow approximately the same distribution as samples from $$\text{Dir}([\alpha_1, \alpha_2, \cdots, \alpha_{k-1}])$$

### My progress so far:

As a sanity check I started with simulations using Numpy. Sampling from $$\text{Dir}([\alpha_1, \alpha_2, \cdots, \alpha_{k-1}, 10^{-10}])$$ and $$\text{Dir}([\alpha_1, \alpha_2, \cdots, \alpha_{k-1}])$$ seems very similar indeed.

Then I've tried substituting the new $$([\alpha_1, \alpha_2, \cdots, \alpha_{k-1}, \epsilon])$$ vector into the expression for the PDF of the Dirichlet distribution. The resulting expression seems very similar to the PDF of Dir$$([\alpha_1, \alpha_2, \cdots, \alpha_{k-1}])$$, but it's being multiplied by $$\frac{1}{x_k\cdot\Gamma(\epsilon)}$$. As $$\epsilon$$ decreases the whole expression tends to zero. Not really sure how to proceed from here.