A Law of Large Numbers Without Replacement Let $(n_1,...,n_r)$ be $r$ positive integers, and let $n=n_1+...+n_r$.
Fo each positive integer $m$ consider an urn containing $mn$ balls,
of which $mn_1$ are of type 1,..., $mn_r$ of type r. For each integer $k$, with $1 \leq k \leq mn$, let $X^{(m,k)}=(X^{(m,k)}_1,...,X^{(m,k)}_r)$, where $X_i$ is the number of
balls of type $i$ drawn from the urn, when $k$ balls are drawn without replacement.
$X^{(m,k)}$ is a random variable with a hypergeometric distribution, and we have
\begin{equation}
E\left(\frac{X^{(m,k)}}{k}\right)=\mu,
\end{equation}
where $\mu_i=n_i/n$. I guess that the following version of the "law of large numbers"
should hold. For any $\delta > 0$ and $\epsilon > 0$, there exists a positive integer
$N$ such that
\begin{equation}
P\left(\left|\left| \frac{X^{(m,k)}}{k} - \mu \right|\right| \leq \delta \right) \geq 1 - \epsilon,
\end{equation}
for all integers $m, k$ such that $N \leq k \leq mn$. Do you have any idea of a possible proof?
Thank you very much for your attention.
Note. This question is related to my other question "Multinomial vs Hypergeometric Distribution: Shapley's Uniformly Better Principle". Actually if the inequality of
this principle would turn out to be true, then a proof of the "law of large numbers"
I conjectured would immediately follow from the proof of the weak law of large numbers for Bernoulli trials (as given in Feller, Introduction to Probability and Its Applications, vol. I).
 A: I finally found the proof. The key element is the inequality of Hoeffding-Chvatal (see e.g. Chvatal, The Tail of the Hypergeometric Distribution, Discrete Mathematics 25(3),285-287,1979 or the online paper by Skala, Hypergeometric Tail Inequalities Ending an Insanity) which is the analogous for the univariate hypergeometric distribution of the Chernoff inequality for the binomial distribution. For each $i=1,...,r$ this inequality writes
(note that $X^{(m,k)}_i$ has a univariate hypergeometric distribution) 
\begin{equation}
P \left( \left| \frac{X^{(1,k)}_1}{k} - \mu_1 \right| \geq \delta \right) \leq
2e^{-2 \delta^2 k},
\end{equation}
so that we get the bound
\begin{equation}
P \left( \left| \left| \frac{X^{(m,k)}}{k} - \mu \right| \right| \geq \delta \right) \leq
2r e^{-2 \delta^2 k}.
\end{equation}
Thank you very much for your attention.
A: A quick thought on what might be an easier method: First use the fact that in the large $m$ limit, the hypergeometric distribution converges to the multinomial distribution. Then use the law of large numbers as normal, since draws from the multinomial distribution are i.i.d.
