Integral limit theorems on large deviations for multidimensional hypergeometric distribution

Integral large deviation theorems are obtained for multidimensional hypergeometric distribution. These theorems allow us to evaluate the probabilities of large deviations with the remainder term of order $O(1/N)$. The corresponding hypergeometric distribution of a random vector $(\mu_1,\dots,\mu_s)$ has the form
$$ \mathbf{P}\{(\mu_1,\dots,\mu_s)=(k_1,\dots,k_s)\}=\frac{C_{M_1}^{k_1}\dotsb C_{M_s}^{k_s}}{C_N^n}\,, $$
and $k_j\le M_j$, $j=1,\dots,s$; 0 in the remaining cases.