A uniform asymptotic expansion for weighted sums of exponentials

We consider the random variable Zn,α=Y1+2αY2+⋯+nαYnZn,α=Y1+2αY2+⋯+nαYn, with α∈Rα∈R and Y1,Y2,…Y1,Y2,… independent and exponentially distributed random variables with mean one. The distribution function of Zn,αZn,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(Zn,α