A series concerning subcritical multitype branching processes (Q756867)

This paper considers a multitype, subcritical, Galton-Watson process. Let \(\rho\) be the maximal eigenvalue of the mean matrix, with associated eigenvector v, e be a vector of ones and \(f_ n(s)\) be the n-th iterate of the offspring generating function. It is known that an `x log x' condition is necessary and sufficient for \(\gamma_ n(s):=\rho^{- n}v\cdot (e-f_ n(s))\) to decrease to \(\gamma (s)>0\). Here it is shown that an `x log\({}^ 2 x'\) condition is necessary and sufficient for \(\sum_{n}(\gamma_ n(s)-\gamma (s))\) to be finite.