Convergence conditions for weighted branching processes

We consider some aspects of the weighted branching processes and, in particular, consider convergence conditions for the processes being a certain analogue of the well-known condition on $X\ln X$ for ordinary branching processes and branching random walks on $\mathbb R$.
The research was supported by the Russian Foundation for Basic Research, grant 99–01–00012, and by RFBR–NNIO, grant 98–01–04132.