Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Malvina Vamvakari

TL;DR

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 is provided.

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.

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

TL;DR

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 is provided.

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.

Paper Structure

This paper contains 3 sections, 2 theorems, 25 equations.

Table of Contents

  1. Brief Introduction
  2. Preliminaries
  3. Main Results

Key Result

Theorem 2

Let $\theta_1=\theta_{1,n}=q^{-\alpha_1 n}$ and $\theta_2=\theta_{2,n}=q^{-\alpha_2 n}$ with $0<a_1,\, a_2<1$ constants and $0<q<1$. Then, for $n \rightarrow \infty$, the $q$-trinomial distribution of the first kind is approximated by a deformed standardized bivariate continuous Stieltjes-Wigert dis where $\mu_{[X_{1}]_{1/q}}$ and $\sigma^2_{[X_{1}]_{1/q}},$ given in meanvarYEb2, are the mean valu

Theorems & Definitions (3)

  • Theorem 2
  • Theorem 4
  • Remark 5