Let $\Sigma_k$ be the $k$-dimensional simplex $\{x_1,\dots x_{k+1}| \sum_j x_1=1\}$. Given a set of parameters $\vec{q}=(q_1, \dots q_{k})$ in $\Sigma_{k-1}$ and a bunch of non-negative integers $\vec{n}=(n_1,\dots, n_{k})$ one can define the multinomial distribution \begin{equation} p(\vec{n};\vec{q})=\frac{(n_1+\dots+n_{k})!}{n_1!\dots n_{k}!}q_1^{n_1}\dots q_{k}^{n_k}\;. \end{equation}
I want to define a probability distribution on the simplex of the fractions $x_i=n_i/N$, where $N=\sum_j n_i$.
Of course for finite $N$ this will be a discrete distribution on $\Sigma_{k-1}$, just like $p(\vec{n};\vec{q})$. But somehow, in some appropriate limit, this will be a well defined continuous probability distribution. Now my question is how to specify "somehow", i.e. how to take this limit and find the explicit limiting distribution.
My plan was to find an $\epsilon(N)$ for a given $N$, which goes to zero as $N\to\infty$ and consider $\epsilon$-balls around a given point $\vec x$, denoted by $B_\epsilon(\vec x)$. We define the set $S_\epsilon(\vec x):=\{\vec n\in \mathbb N^k|\frac{n_i}{N}\in B_\epsilon(x_i)\;\;\forall i\}$ and then take the limit: \begin{equation} \tilde p(\vec x;\vec q):=\limsup_{N\to \infty}\sum_{\vec n\in S_{\epsilon(N)}(\vec x)}p(\vec n;\vec q) \end{equation}
I think this is somehow related to the Dirichlet distribution. Maybe someone can enlighten me. I have also posted a related question here: https://math.stackexchange.com/questions/1401223/probability-distribution-on-the-simplex-with-support-on-the-faces . However I guess that the limit, as described here, will have no support on th faces of the simplex.
Nice idea, but unfortunately, since both the means and the variances grow with $N$, the distribution is increasingly concentrated near the expected values of the fractions, $x_i=q_i$, so the limiting distribution is the delta distribution at that point – not terribly interesting, unfortunately.