Variations of Central Limit Theorems and Stirling numbers of the First Kind

TL;DR

This paper introduces a new parametrization of sequences related to Stirling numbers of the first kind, proving central and local limit theorems for each parameter, extending classical results and providing various applications.

Contribution

It develops a novel parametrization connecting binomial coefficients and Stirling numbers, and proves limit theorems for these sequences, extending classical asymptotic normality results.

Findings

Established a new parametrization of sequences between binomial coefficients and Stirling numbers.

Proved central limit theorems for each parameter in the new sequence.

Extended classical asymptotic normality results to a broader class of sequences.

Abstract

We construct a new parametrization of double sequences $\{A_{n,k}(s)\}_{n,k}$ between $A_{n,k}(0)= \binom{n-1}{k-1}$ and $A_{n,k}(1)= \frac{1}{n!}\stirl{n}{k}$, where $\stirl{n}{k}$ are the unsigned Stirling numbers of the first kind. For each $s$ we prove a central limit theorem and a local limit theorem. This extends the de\,Moivre--Laplace central limit theorem and Goncharov's result, that unsigned Stirling numbers of the first kind are asymptotically normal. Herewith, we provide several applications.