A central limit theorem for numbers satisfying a class of triangular arrays
type: Item
instance of: scholarly article
title: A central limit theorem for numbers satisfying a class of triangular arrays
zbMATH Open document ID: 0542.60026
DOI: 10.1016/0012-365X(84)90022-0
published in: Discrete Mathematics
publication date: 1984-01-01T00:00:00Z
review text: With a double sequence of non-negative numbers, say A(m,n), we associate the combinatorial distribution \(P_ m(n)=A(m,n)\lambda^ n/A_ m(\lambda)\), \(n=0,1,...,m\), \(\lambda>0\), \(A_ m(\lambda)=\sum^{m}_{j=0}A(m,j)\lambda^ j\). We say that A(m,n) satisfy a central limit theorem or they are asymptotically normal with mean \(\mu_ m\) and variance \(\sigma^ 2_ m\) if \[ \lim_{m\to \infty}\sup_{x}| \sum_{n\leq \sigma_ mx+\mu_ m}p_ m(n)- (2\pi)^{-1/2}\int^{x}_{-\infty}\exp [-t^ 2/2]dt| =0. \] In this paper we suppose that A(m,n) satisfy the following class of triangular arrays \[ A(m+1,n)=[\alpha_ 1(m)n+\alpha_ 0(m)]A(m,n)+[b_ 1(m)n+b_ 0(m)]A(m,n-1), \] \[ n=0,1,...,m+1,\quad m=0,1,...,\quad if\quad \alpha_ 0(0)\neq 0;\quad n=1,2,...,m+1,\quad m=0,1,...,\quad if\quad \alpha_ 0(0)=0, \] with boundary conditions \(A(0,0)=1\), \(A(0,n)=0\) if \(n\neq 0\), and with \(\{\alpha_ 1(m)n+\alpha_ 0(m)\}\) and \(\{b_ 1(m)n+b_ 0(m)\}\) positive for the corresponding values of n and m and we prove that they are asymptotically normal when \(\alpha_ 0(m)\), \(b_ 0(m)\) are linear functions in m and when \(\alpha_ 1(m)\), \(b_ 1(m)\), \(\alpha_ 0(m)\), \(b_ 0(m)\) are ''multiples'' of a common function of m. Moreover, this work covers in particular many known results as for A(m,n) the Stirling numbers of the second kind or the Eulerian numbers, etc. and most standard cases of triangular arrays.
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60C05
zbMATH DE Number: 3862132
zbMATH Keywords: combinatorial distribution
zbMATH Keywords: central limit theorem
zbMATH Keywords: asymptotically normal
zbMATH Keywords: triangular arrays
author: Andreas Kyriakoussis
MaRDI profile type: Publication
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full work available at URL: 0012-365x(84)90022-0
OpenAlex ID: W2067848606
description: scientific article; zbMATH DE number 3862132