Normal convergence by higher semi-invariants with applications to sums of dependent random variables and random graphs. (English) Zbl 0639.60029

Janson, Svante

Ann. Probab. 16, No. 1, 305-312 (1988).

Let \((X_ n)_{n\in {\mathbb{N}}}\) be a sequence of real r.v. with semiinvariants \(\kappa_ j(X_ n)\), and let \(N(\mu,\sigma^ 2)\) be a normally distributed r.v. with mean \(\mu\in {\mathbb{R}}\) and variance \(\sigma^ 2>0.\)
The author shows that if \(\kappa_ 1(X_ n)=E(X_ n)\to \mu\), \(\kappa_ 2(X_ n)=var(X_ n)\to \sigma^ 2\), and \(\kappa_ j(X_ n)\to 0\) for every \(j\geq m\), \(m\geq 3\) fixed, \(n\to \infty\), then \(X_ n\) tends in distribution to \(N(\mu,\sigma^ 2)\), \(n\to \infty\). No information is needed on the remaining semiinvariants \(\kappa_ j(X_ n)\), \(3\leq j<m\), \(n\in {\mathbb{N}}\), but to define the semiinvariants, \(E(| X_ n|^ j)<\infty\), \(j\in {\mathbb{N}}\), \(n\geq n_ j.\)
This result is applied to give a new criterion for asymptotic normality of sums of dependent variables. An example is included where this criterion is applied to the number of induced subgraphs of a particular type in a random graph.

Reviewer: L.Hahn

MSC:

60F05	Central limit and other weak theorems
05C80	Random graphs (graph-theoretic aspects)

Keywords:

central limit theorem; semiinvariants; criterion for asymptotic normality of sums of dependent variables; random graph