A uniform asymptotic expansion for weighted sums of exponentials

作者：van Leeuwaarden J S H^*; Temme N M

来源：Statistics & Probability Letters, 2011, 81(11): 1571-1579.

DOI：10.1016/j.spl.2011.05.013

• Asymptotic analysis
• Saddle point approximation
• Exponential random variables
• Gumbel distribution
• Kolmogorov distribution

We consider the random variable Z(n,alpha) = Y(1) + 2(alpha)Y(2) + ... + n(alpha)Y(n), with alpha is an element of 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 P(Z(n,alpha) < x) that remains valid inside (alpha >= -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 Kingman and Volkov (2003).

• 出版日期2011-11