the generating functions $F_{1}(z; t, x,y) $ and $ F (z; t, x,y) $ are continuously differentiable with respect to $t$ uniformly in $ x,y\in \mathbb{Z}^d$, satisfy the inequalities $ 0\leqslant F_{1}(z; t,x,y), F (z; t, x,y) \leqslant 1 $ and the backward Kolmogorov differential equation .

By Lemma \[L1\] we obtain the assertion of the theorem for $F_{1}(z; t, x)$ and $F_{1}(z; t,x, y)$ and by [@YarBRW:r Lemma 1.2.1] for $F(z; t, x)$ and $F(z; t,x, y)$.

Theorem \[L:osn\] implies that for each $n\geq 1$ the bahaviour of $m_{n}(t,x):=E\mu_{x,t}^{n}$ in the model described in [@YarBRW:r] coincides with the behaviour of $M_{n}(t,x)$ in the model under consideration, and for each $n\geq 1$ the behaviour of $m_{n}(t,x,y):=E\mu_{x,t}^{n}(y)$ in the model described in [@YarBRW:r] coincides with the behaviour of $M_{n}(t,x,y):=E\eta_{x,t}^{n}(y)$. This demonstrates that the results obtained for BRWs with one initial particle may be applied for the study of BRWs with infinitely many initial particles. Owing to Theorem \[L:osn\] we can rewrite the limit theorem, proved in [@YarBRW:r] for $M_{n}(t, x, y)$ and $ M_{n}(t, x)$. Recall [@YarBRW:r] that the intensity value $\beta_{c}$ is called *critical* if for $\beta<\beta_{c}$ all the moments tend to zero or bounded on infinity, while for $\beta>\beta_{c}$ 