We consider the random variable $Z_{n,\alpha}=Y_1+2^\alpha Y_2 +\ldots + n^\alpha Y_n$, with $\alpha\in\mathbb{R}$ and $Y_1,Y_2,\ldots$ 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 $\mathbb{P}(Z_{n,\alpha}<x)$ that remains valid inside ($\alpha\geq -1/2$) and outside ($\alpha< -1/2$) 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 \cite{Kingman:2004:OK}.

Additional Metadata
Keywords Saddle point approximation, exponential random variables, Gumbel distribution, Kolmogorov distribution
MSC Central limit and other weak theorems (msc 60F05)
Publisher North-Holland
Journal Statistics and Probability Letters
van Leeuwaarden, J, & Temme, N.M. (2011). A uniform asymptotic expansion for weighted sums of exponentials. Statistics and Probability Letters, 81(11), 1571–1579.