A uniform asymptotic expansion for weighted sums of exponentials

Abstract

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 [7].

On sequential selection and a first passage problem for the Poisson process

“…Plugging the lower bound (19) in (23) yields the bound σ 2 (t) < √ 2t + 1 − 1, which for large t is too far from (yet to be justified) (15). But working other way round we substitute (15) with indefinite smaller order remainder in (23), and work out the quadratic equation to extract the value of the sought limit constant:…”

On the Estimation of Parameter of Weighted Sums of Exponential Distribution

“…Because 𝛼 = −2 the random variable 𝑍 𝑛,𝛼 is Kolmogorov distribution. In general case, the distribution function is a series with alternating sign that Van Leeuwarden and Temme [5] derived an approximation uniform expansion for P (𝑍 𝑛,𝛼 < 𝑥) by applying an extended version of the saddle point method.…”