Inverse relationship and limiting forms of lagrange distributions

The basic Lagrange distribution is shown to be inversely related to some distribution obtained from the arrival distribution. Using this inverse relationship and the method of moments, we show that Lagrange distributions converge to normal distributions under some conditions and to inverse Gaussian distributions under other conditions. Although Consul and Shenton (1973) gave theorems about the limiting forms of Lagrange distributions, they implicitly assume several conditions and their theorems do not hold in general. We give counterexamples to Consul and Shenton's theorems and examples of convergence to normal and to inverse Gaussian distributions.