Local Limit Theorems for $q$-Multinomial and Multiple Heine Distributions

TL;DR

This paper proves local limit theorems showing how q-multinomial and multiple Heine distributions converge to a multivariate Stieltjes-Wigert type distribution, enhancing understanding of their asymptotic behavior.

Contribution

It establishes the first local limit theorems for q-multinomial and multiple Heine distributions, detailing their convergence to a specific multivariate distribution.

Findings

Pointwise convergence of q-multinomial distribution to a Stieltjes-Wigert type distribution

Convergence of multiple Heine distribution to the same limit

Provides a rigorous basis for asymptotic analysis of these distributions

Abstract

In this work we establish local limit theorems for q-multinomial and multiple Heine distributions. Specifically, the pointwise convergence of the q-multinomial distribution of the first kind, as well as for its discrete limit, the multiple Heine distribution, to a multivariate Stieltjes-Wigert type distribution, are provided.