Let $X_1,\dots,X_N$ denote a collection of independent samples of a uniform multinomial random variable in $\mathbb{Z}^k$, with the number of trials equal to $n\ll k$. (By "uniform", I mean we have success probability $p_i=1/k$ for all $i\in\{1,\dots,k\}$). Since $n\ll k$, we can safely assume that $X_i\in \{0,1\}^k$ for all $X_i$ (or at least the vast majority of them). Let $Y_i \in \mathbb{Z}^k$ be obtained by deleting $m$ of the nonzero entries of $X_i$. There are obviously $\binom{n}{m}^N$ possible collections $Y_1,\dots,Y_N$ that can be realized from $X_1,\dots,X_N$.
I am interested in the possible values of the sample average $$\frac{1}{ (n-m)N}\sum_{i=1}^N Y_i$$ for all choices of $Y_1,\dots,Y_N$. Certainly, the vast majority of such averages are roughly equal to $( \frac{1}{k},\dots,\frac{1}{k} )$. However, for a fixed probability vector $\mathbf{p}\in\mathbb{R}^k$, can we say anything about the number of collections $Y_1,\dots,Y_N$ whose sample average is "roughly" $\mathbf{p}$, as $N\to\infty$?
