The asymptotic central-raw ratios for binomial, Poisson and normal distributions

The n-th central-raw ratios r(n)(xi) = E(xi-E xi)(n) /E xi(n) of random variable xi with non-zero expectation are considered in this article. Hayman's method for expansion of the entire function is used to derive the asymptotic formulas of the n-th raw and central moments for binomial and Poisson distributions. Laplace's expansion of integral is used to obtain the asymptotic formula of the n-th raw moment for normal distribution. Consequently, the n-th centralraw ratios are shown to be infinitesimal with different patterns for binomial B(N, p), Poisson P(lambda) and normal N(mu, sigma(2)) distributions, respectively.

Automated summary

This article investigates the n-th central-raw ratios r(n)(xi) of random variable xi with non-zero expectation. It uses Hayman's method to derive asymptotic formulas for the n-th raw and central moments for binomial and Poisson distributions, and Laplace's expansion to obtain the asymptotic formula for the n-th raw moment of the normal distribution. As a result, the n-th central-raw ratios show different patterns for binomial B(N, p), Poisson P(lambda), and normal N(mu, sigma(2)) distributions.