The supercritical Galton-Watson process in varying environments -- Seneta-Heyde norming
The authors consider a Galton-Watson process in varying environment \(\{Z_ n\}\) defined by \(Z_ 0=1\), \(Z_{n+1}=\sum^{Z_ n}_{i=1} X_{n,i}\), \(n \geq 0\), and \(\{X_{n,i;i}\}\) are independent identically distributed copies of a random variable \(X_ n\). A sufficient condition is given for \(\{Z_ n\}\) to have exactly one rate of growth, in the sense that there is a sequence of constants \(\{C_ n\}\) such that \(Z_ n/C_ n\) converges to a finite limit that is strictly positive on the survival set, \(\{Z_ n \to \infty\}\). The condition is that the process is uniformly supercritical and that \(\{X_ n/ \mu_ n\}\) is dominated by a random variable \(X\) with \(EX<\infty\), \(\mu_ n\) denoting \(EX_ n\). A construction is given for \(\{C_ n\}\).
scientific article; zbMATH DE number 493306
scientific article