On an Asymptotic Series of Ramanujan

Details On an Asymptotic Series of Ramanujan On an Asymptotic Series of Ramanujan

2009-04-07

arxiv.org/abs/0904.1226v1

Ramanujan J. 20 (2009) 179--188

Mathematics

Statistics Theory

To appear, Ramanujan J

Scientific paper

10.1007/s11139-009-9169-x

An asymptotic series in Ramanujan's second notebook (Entry 10, Chapter 3) is concerned with the behavior of the expected value of $\phi(X)$ for large $\lambda$ where $X$ is a Poisson random variable with mean $\lambda$ and $\phi$ is a function satisfying certain growth conditions. We generalize this by studying the asymptotics of the expected value of $\phi(X)$ 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.

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