TY - JOUR

T1 - A uniform asymptotic expansion for weighted sums of exponentials

AU - Leeuwaarden, van, J.S.H.

AU - Temme, N.M.

PY - 2011

Y1 - 2011

N2 - 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).

AB - 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).

U2 - 10.1016/j.spl.2011.05.013

DO - 10.1016/j.spl.2011.05.013

M3 - Article

VL - 81

SP - 1571

EP - 1579

JO - Statistics and Probability Letters

JF - Statistics and Probability Letters

SN - 0167-7152

IS - 11

ER -