Two-dimensional renewal theorems with weak moment conditions and critical Bellman - Harris branching processes

We consider critical Bellman Harris processes with two types of particles. The tail of the lifetime distribution of the first type particles decreases as 0(t(-2)), the tail of the lifetime distribution of the second type particles is regularly varying with the index in (-1, 0). Such processes are connected with the matrix renewal functions of special two-dimensional renewal processes. V. A. Vatutin and the author have used the asymptotics of these matrix renewal functions and their first and second order increments in the proofs of several limit theorems for branching processes. Here we investigate the properties of such matrix renewal functions under significantly weaker conditions on the lifetime distributions and apply the results to the description of the asymptotics of several moments of branching processes and of their increments.