Let $0<\beta\leq 1$, $\theta\geq0$, and consider the series $$ S(\theta,\beta)=\sum_{k=0}^\infty\left(\frac{e^{-\theta}\theta^k}{k!}\right)^\beta. $$ How might we determine the upper and lower indicies $k_\ell$ and $k_u$ such that $$ \sum_{k=k_\ell}^{k_u}\left(\frac{e^{-\theta}\theta^k}{k!}\right)^\beta=(1-\epsilon)S(\theta,\beta) $$ for choice of small $\epsilon>0$?
The case for $\beta=1$ is easy since the terms of the series represent Poisson probabilities. This means that one may use the Poisson quantile function, after specifying $\epsilon$, to write $$ k_\ell(\theta,1)=F^{-1}_\theta(\epsilon/2) $$ and $$ k_u(\theta,1)=F^{-1}_\theta(1-\epsilon/2). $$ For $\beta<1$ the terms in the series, if interpreted as probabilities after multiplying a normailzation constant, result in a probability distribution that is of larger variance and so I suspect $$ k_\ell(\theta,\beta)\leq k_\ell(\theta,1) $$ and $$ k_u(\theta,\beta)\geq k_u(\theta,1). $$ For my particular application, the less terms in the series I need to obtain the approximation the better. Computational cost of adding additional terms will be expensive.
I shall derive approximations for $k_{\ell ,u} (\theta ,\beta )$ when $\theta$ is large. Assume that $\beta>0$ is fixed. By the results of this paper, we have $$ S(\theta ,\beta ) \sim \frac{1}{{(2\pi \theta )^{(\beta - 1)/2} \sqrt \beta }} $$ as $\theta \to +\infty$. By this answer, we have $$ (\theta + t\sqrt \theta )! \sim \sqrt {2\pi }\, \theta ^{\theta + t\sqrt \theta + 1/2} \exp \left( { - \theta + \frac{{t^2 }}{2}} \right) $$ as $\theta \to +\infty$ uniformly for bounded values of $t$. Thus, \begin{align*} \sum\limits_{k = \theta - \tau \sqrt \theta }^{\theta + \tau \sqrt \theta } {\left( {\frac{{{\rm e}^{ - \theta } \theta ^k }}{{k!}}} \right)^\beta } & \sim \int_{\theta - \tau \sqrt \theta }^{\theta + \tau \sqrt \theta } {\left( {\frac{{{\rm e}^{ - \theta } \theta ^s }}{{s!}}} \right)^\beta {\rm d}s} = \sqrt \theta \int_{ - \tau }^\tau {\left( {\frac{{{\rm e}^{ - \theta } \theta ^{\theta + t\sqrt \theta } }}{{(\theta + t\sqrt \theta )!}}} \right)^\beta {\rm d}t} \\ & \sim \frac{{\sqrt \theta }}{{(2\pi \theta )^{\beta /2} }}\int_{ - \tau }^\tau {\exp \left( { - \frac{{t^2 }}{2}\beta } \right){\rm d}t} = \frac{1}{{(2\pi \theta )^{(\beta - 1)/2} \sqrt \beta }}\operatorname{erf}\left( {\sqrt {\frac{\beta }{2}} \tau } \right) \\ & \sim S(\theta ,\beta )\operatorname{erf}\left( {\sqrt {\frac{\beta }{2}} \tau } \right) = S(\theta ,\beta )\left(1-\operatorname{erfc}\left( {\sqrt {\frac{\beta }{2}} \tau } \right)\right) \end{align*} for large positive $\theta$ and bounded $\tau>0$. Here $\operatorname{erf}$ and $\operatorname{erfc}$ denote the error function and the complementary error function, respectively. Thus, for any fixed $0<\varepsilon<1$, one may take $$ k_{\ell ,u} (\theta ,\beta ) \approx \theta \mp \tau _\varepsilon \sqrt \theta $$ where $\tau_\varepsilon$ is the unique positive solution of the equation $$ \operatorname{erfc}\left( {\sqrt {\frac{\beta }{2}} \tau _\varepsilon } \right) = \varepsilon . $$ (See here for approximate solutions.)