A uniform asymptotic expansion for weighted sums of exponentials

J.S.H. van Leeuwaarden (1) , N.M. Temme

1 Department of mathematics and computing science [Eindhoven]

We consider the random variable , with and independent and exponentially distributed random variables with mean one. The distribution function of 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 that remains valid inside () and outside () the domain of attraction of the central limit theorem. We discuss several special cases, including , for which we sharpen some of the results in Kingman and Volkov (2003).

Mots clés: 60F05; 33B20; 41A60; Asymptotic analysis; Saddle point approximation; Exponential random variables; Gumbel distribution; Kolmogorov distribution; OK Corral model

Domaines: Bio-informatique [q-bio.QM]; Bio-Informatique, Biologie Systémique [q-bio.QM]

J.S.H. van Leeuwaarden, N.M. Temme. A uniform asymptotic expansion for weighted sums of exponentials. Statistics and Probability Letters, 2011, 81 (11), pp.1571. ⟨10.1016/j.spl.2011.05.013⟩. ⟨hal-00789868⟩