A necessary and sufficient condition for a branching process in a random environment to grow like the product of its means
Let \(\{Z_ n\}\) be a branching process in a stationary ergodic sequence of environments. Let \(F_ n\) be the distribution of the family size to members of the n th generation, with mean \(m_ n\), conditional on the environmental sequence. The supercritical case, when \(E(\log (m_ 0))>0\), is mainly considered here. Let \(\pi_ n=m_ 0m_ 1...m_{n-1}\), then it is shown that the limit, W, of the martingale \(Z_ n/\pi_ n\) is non-degenerate exactly when \(\sum (m_ n)^{-1}\int x I(x>\pi_{n+1})F_ n(dx)\) is finite almost surely (provided a minor technical condition holds). As a consequence the finiteness of \(E(Z_ 1\log\) \(+(Z_ 1)/m_ 0)\) suffices for non- degeneracy, and this is also necessary if the environments are i.i.d.
scientific article; zbMATH DE number 4048801
scientific article