On an Asymptotic Series of Ramanujan

Abstract

An asymptotic series in Ramanujan's second notebook (Entry 10, Chapter 3) is concerned with the behavior of the expected value of $\varphi (X)$ for large $\lambda$ where $X$ is a Poisson random variable with mean $\lambda$ and $\varphi$ is a function satisfying certain growth conditions. We generalize this by studying the asymptotics of the expected value of $\varphi (X)$ when the distribution of $X$ belongs to a suitable family indexed by a convolution parameter. Examples include the problem of inverse moments for distribution families such as the binomial or the negative binomial.

To appear, Ramanujan J

References in corpus (1)

• Asymptotic expansion for inverse moments of binomial and Poisson distributions

Cited by in corpus (1)

• Sharp Bounds on the Entropy of the Poisson Law and Related Quantities