A uniform asymptotic expansion for weighted sums of exponentials

We consider the random variable Z n , α = Y 1 + 2 α Y 2 + ⋯ + n α Y n , with α ∈ R and Y 1 , Y 2 , … independent and exponentially distributed random variables with mean one. The distribution function of Z n , α 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 P ( Z n , α < x ) that remains valid inside ( α ≥ − 1 / 2 ) and outside ( α < − 1 / 2 ) the domain of attraction of the central limit theorem. We discuss several special cases, including α = 1 , for which we sharpen some of the results in Kingman and Volkov (2003).