On large deviations in the Poisson approximation

This paper proves a general lemma comparing the behavior of probabilities of large deviations $\mathbf{P}(X\ge x)$ of a random variable $X$ against the Poisson distribution $1-P(x,\lambda)$ ($\lambda$ is the parameter of the Poisson distribution). When upper bounds are known for the factorial cumulants $\widetilde\Gamma_k (x)$ of $k$th order:
$$ |\widetilde\Gamma _k (X)|\le\frac{k!\lambda}{\Delta^{k-1}}\quad\text{for }\forall k\ge2 $$
for some $\Delta>0$, then large deviations may be compared in the interval $1\le x-\lambda<\delta\lambda\Delta$, $0<\delta<1$. For such $x$
$$ \frac{\mathbf{P}(X\ge x)}{1-P(x,\lambda)}=e^{L(x)}\biggl(1+\theta_1\frac{1+\lambda}{x}+\theta_2\frac{(x-\lambda)^{3/2}}{\Delta}\biggr), $$
where $L(x)$ is a power series and $|\theta_i|<C(\delta)$, $i=1,2$.