# Limit of the multinomial distribution on the simplex

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.
• I'd like to point out that concentrating to a delta function can happen but doesn't have to, just like how the discrete binomial distribution Bin$(n,\, p)$ can also approach the boring delta distribution when scaled "wrongly" by $n$ and not by the "correct" $\sqrt{n}$ that yields its continuous version of Gaussian. I believe there's an appropriate scale of convergence in the $k$ dimensional space. I don't know how to work out the details for now. – Lee David Chung Lin Sep 28 '16 at 7:36
• A naive guess would be $N^{1/k}$, where this $N$ and $k$ are the same notation as OP. – Lee David Chung Lin Sep 28 '16 at 7:43