On the asymptotics of the probabilities of large deviations for a negative polynomial distribution

We consider the polynomial scheme of trials with outcomes $E_0,E_1,\dots,E_N$ and the corresponding probabilities $p_0,p_1,\dots,p_N$. We assume that the trials are performed until the $r$th occurrence of the outcome $E_0$, $r=1,2,\dotsc$ If $\eta_j(r)$ is the number of occurrences of the outcome $E_j$ at the stopping time, $j=1,\dots,N$, and $\eta(r)=(\eta_1(r),\dots,\eta_N(r))$, then the vector $\eta(r)$ has the negative polynomial distribution. Under the assumptions that $N\in\mathbf N$ and the positive probabilities $p_0,p_1,\dots,p_N$ are fixed, that $r\to\infty$ and $k_1,\dots,k_N\to\infty$ so that the parameters $\beta_j=k_j/r$ satisfy the inequalities $\beta_j\ge\varepsilon$, where $\varepsilon$ is a positive constant, $j=1,\dots,N$, and under some additional constraints, we give asymptotic estimates of the probabilities of large deviations
$$ \mathsf P\{\eta_j(r)\le k_j,\ j=1,\dots,N\}, \qquad \mathsf P\{\eta_j(r)\ge k_j,\ j=1,\dots, N\}. $$
In order to derive these asymptotic estimates, we use the multidimensional saddle-point method in the form suggested by Good.