A uniform asymptotic expansion for weighted sums of exponentials

We consider the random variable $Z_{n,\alpha}=Y_1 + 2^\alpha Y_2 +...+ n^\alpha Y_n$, with $\alpha \in \mathrm{R}$ and $Y_1,Y_2,...$ independent and exponentially distributed random variables with mean one. The distribution function of $Z_{n,\alpha}$ is in terms of a series with alternating signs, causing great numerical difficulties. Using an extended version of the saddle point method, we derive a uniform asymptotic expansion for $\mathrm{P} (Z_{n,\alpha} <x)$ that remains valid inside ($\alpha \geq -\frac{1}{2}$) and outside ($\alpha <-\frac{1}{2}$) the domain of attraction of the central limit theorem. We discuss several special cases, including $\alpha=1$, for which we sharpen some of the results in Kingman and Volkov (2003).