The ubiquitous Gaussian limit law in analytic combinatorics

Data is everywhere, notably in this information explosion era. Beyond the first-order summary of a sample by its average value or its median, the bell-shaped Gaussian (or normal) curve has long been observed in the histograms of many data samples since the early history of probability, statistics and related fields. The Gaussian distribution, known commonly under the name of Law of Frequency of Errors, first appeared in 1733 in de Moivre’s works (see his Doctrine of Chance, second edition, first published in 1738, and [9]) as the limit of suitably normalized binomial distributions; see Figure 1. The following intuitive arguments of G. Galilei (see [9, Ch. 1]) may be regarded as the first-level description of the Law: