Convergence of critical multitype Galton-Watson branching processes
Let \(Z_ n\) denote a positive regular, non-singular, d-type, critical Galton-Watson process with not necessarily finite second moments of its offspring probability generating function F(s). Define \(a_ n=v(1-F_ n(0))\), when \(F_ n\) denotes the n-th iterate of F and v the normalized left eigenvector corresponding to the (maximal) eigenvalue 1 of the matrix of the offspring expectations. Let u denote the corresponding right eigenvector. Considering \(Z_{[nt]}\) as a random element in the appropriate function space the author can prove two diffusion type limit theorems by increasing the initial population size if all the second moments of \(Z_ 1\) are finite: Firstly, not conditioning on non-extinction for \(\hat Y_ n(t)=a_ nZ_{[nt]}u\) with \(\hat Y_ n(0)=x_ 0+o(1)\) and, secondly, conditioning on non-extincting for \(Y_ n(t)=a_ nZ_{[nt]}u| Z_ n\neq 0\) with \(Z_ 0=a_ n^{-1}x_ 0v+o(a_ n^{-1}).\) This generalizes results of \textit{T. Lindvall} [J. Appl. Probab. 9, 445- 450 (1972; Zbl 0238.60063), \textit{J. Lamperti} and \textit{P. Ney} [Teor. Veroyatn. Primen. 13, 126-137 (1968; Zbl 0253.60073)] and \textit{T. Nakagawa} [J. Multivariate Anal. 12, 161-177 (1982; Zbl 0492.60081)] and is obtained by convergence theorems for the finite-dimensional distributions under Vatutin's infinite variance condition.
convergence theorems for the finite-dimensional distributions
scientific article; zbMATH DE number 3984268