On an Asymptotic Series of Ramanujan

TL;DR

This paper explores the asymptotic behavior of expected values of functions of random variables, extending Ramanujan's series to broader distribution families like binomial and negative binomial.

Contribution

It generalizes Ramanujan's asymptotic series to a wider class of distributions, including convolution families such as binomial and negative binomial.

Findings

Derived asymptotic formulas for inverse moments

Extended Ramanujan's series to new distribution families

Provided examples with binomial and negative binomial distributions

Abstract

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.