Let $C=C_{n,m}=(c_1,\ldots,c_n)$ be a random weak $n$-part integer composition of $m$ drawn uniformly from all $\binom{m+n-1}m$ such compositions. Suppose $m\to\infty$ as $n\to\infty$ (e.g. $m\sim n^\alpha$ for some $\alpha>0$). Given fixed $k>1$, let $C_k=(c_1,\ldots,c_{\lfloor n/k\rfloor})$ and $S_k$ be the sum of its terms.
Is there a short proof (or does it follow from a more general result) that, as $n\to\infty$, $ \textrm{Prob}\big[ |S_k/m-1/k|>\varepsilon \big] \to 0 $?
As I understand it, it is sufficent for a LLN to hold for $\mathrm{Cov}[c_i,c_j]$ to be nonpositive (is there a standard reference for this?). In this case, it seems evident (combinatorially) that we have $\mathrm{Cov}[c_i,c_j]<0$. Is there a simple proof of that fact?