On an elementary ratio technique for proving convergence of known distributions

Abstract

Proschan offered a simple informal, the so called, ratio technique for deriving the convergence of binomial $B(n,p)$ distribution to a limiting normal as the number $n$ of trials increases to infinity (see also Bagui and Mehra). The technique offers only a heuristic derivation and not a viable proof. The object of the present article is to provide a frame-work so that this heuristic technique gets accepted as a method of proof, under a set of conditions that are readily verifiable.

Keywords:

• Binomial
• first passage
• random walk
• ratio technique

Acknowledgments

The authors would like to thank the Editor, the Associate editor, and the Referee for their many constructive comments and suggestions on the earlier version of the article which led to the present improved version.