A central limit theorem for numbers satisfying a class of triangular arrays
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.
scientific article; zbMATH DE number 3862132