Deviations from typical type proportions in critical multitype Galton–Watson processes

Consider a critical $K$-type Galton–Watson process $\{\mathbf{Z}(t): t=0,1,\ldots \} $ and a real vector $\mathbf{w}=(w_{1},\ldots ,w_{K})^{\top}$. It is well known that under rather general assumptions, $\langle \mathbf{Z} (t),\mathbf{w}\rangle :=\sum_{k}Z_{k}(t)w_{k}$ conditioned on nonextinction and appropriately scaled has a limit in law as $t\uparrow \infty$ [V. A. Vatutin, Math. USSR Sb., 32 (1977), pp. 215–225]. However, the limit degenerates to $\,0$ if the vector $\mathbf{w}$ deviates seriously from "typical" type proportions, i.e., if $\mathbf{w}$ is orthogonal to the left eigenvectors related to the maximal eigenvalue of the mean value matrix. We show that in this case (under reasonable additional assumptions on the offspring laws) there exists a better normalization which leads to a nondegenerate limit. Opposed to the finite variance case, which was already resolved in [K. Athreya and P. Ney, Ann. Probab., 2 (1974), pp. 339–343] and [I. S. Badalbaev and A. Mukhitdinov, Theory Probab. Appl., 34 (1989), pp. 690–694], the limit law (for instance, its “index”) may seriously depend on $\mathbf{w}$.