A Quick Empirical Reproof of the Asymptotic Normality of the Hirsch Citation Index (First proved by Canfield, Corteel, and Savage)

TL;DR

This paper provides a rapid empirical verification of the asymptotic normality of the Hirsch index, offering a practical alternative to complex rigorous proofs and suggesting broader applicability of empirical methods in mathematical proofs.

Contribution

It introduces an empirical approach to verify asymptotic normality, simplifying the process compared to traditional deep analytical proofs.

Findings

Empirical evidence supports the asymptotic normality of the Hirsch index.

The empirical method is faster and easier than rigorous proofs.

Potential applicability to other complex asymptotic problems.

Abstract

Once upon a time there was an esoteric and specialized notion, called "size of the Durfee square", of interest to at most 100 specialists in the whole world. Then it was kissed by a prince called Jorge Hirsch, and became the famous (and to quite a few people, infamous) h-index, of interest to every scientist, and scholar, since it tells you how productive a scientist (or scholar) you are! When Rodney Canfield, Sylvie Corteel, and Carla Savage wrote their beautiful 1998 article proving, rigorously, by a very deep and intricate analysis, the asymptotic normality of the random variable "size of Durfee square" defined on integer-partitions of n (as n goes to infinity), with precise asymptotics for the mean and variance, they did not dream that one day their result should be of interest to everyone who has ever published a paper. However Canfield et. al. had to work really hard to prove their deep result. Here we take an "empirical" shortcut, that proves the same thing much faster (modulo routine number- and symbol- crunching). More importantly, the empirical methodology should be useful in many other cases where rigorous proofs are either too hard, or not worth the trouble!