Dimensions of supercritical branching processes in varying environments (Q1767743)

Let \(Z(n)\) be a branching process in varying environment (BPVE) and \(u_n\) be the expected number of children of individuals of the \(n\)th generation. A BPVE is called uniformly supercritical if there are \(A>0\) and \(c>1\) such that \(\prod_{j=i}^{n+i-1}u_j\geq Ac^n,\;i,n\geq 1.\) Let \(\partial\Gamma\) be the boundary of a family tree \(\Gamma\) of a uniformly supercritical BPVE, and \(\dim_H\partial\Gamma,\) \(\overline{\dim}_B\partial\Gamma,\) \(\dim_P\partial\Gamma\) be the Hausdorff dimension, the upper box dimension and the packing dimension of \(\partial\Gamma,\) respectively. It is shown that \[ \dim_H\partial\Gamma=\liminf_{n\to\infty}n^{-1}\ln EZ(n),\quad \overline{\dim}_B\partial\Gamma=\dim_P\partial\Gamma=\limsup_{n\to\infty}n^{-1}\ln EZ(n). \]