Transient Phenomena in Branching Stochastic Processes

Let $\mu _k (t)=\{\mu _{k_1}(t),\dots,\mu _{kn}(t)\} $ be a branching process with $n$ types of particles and let
$$\mathbf P\left\{\mu _{kj}(t)=\omega_j,j=1,\dots,n\right\}=\delta_k^\omega+p_k^\omega t+o(t),\quad k=1, \dots,n,$$
when $t\to 0$. Here $\omega=\{\omega_1,\dots,\omega_n\},\delta_k^\omega=1$ for $\omega_k=1,\omega_j=0, j\ne k$, and $\delta_k^\omega=0$ in other cases. We define the generating functions by $f_k\left({x_1, \dots x_n}\right)=\sum {p_k^\omega x_1^{\omega_1}\dots x_n^{\omega_n}},k=1,\dots,n,$ and denote factorial moments by
$$a_{kj}=\frac{\partial f_k}{\partial x_j}\biggr|_{x=1},\quad b_{ij}^k=\frac{\partial^2f_k}{\partial x_i\partial x_j}\biggr|_{x=1},\quad c_{ijl}^{(k)}= \frac{\partial^3f_k}{\partial x_i\partial x_j\partial x_l}\biggr|_{x=1}.$$
Let $\mathfrak{A}$ be the compact set of an undecomposable matrix $a=\|{a_{kj}}\|,k,j=1,\dots,n,\lambda=\max_{1\leq i\leq n}(\operatorname{Re} \lambda_i)$, where the numbers $\lambda _i$ satisfy the equality $|{a-\lambda_i E}|=0$ (Ebeing the unity matrix) and let $v=\left\{v_i\right\}_{i=1}^n,u=\left\{u_i\right\}_{i=1}^n$ satisfy the equalities
$$au= \lambda u,\quad va=\lambda v,\quad\sum\limits_{k=1}^n{v_k^2=1,}\quad\sum\limits_{k=1}^n {u_k v_k=1}.$$

Let $\mathrm K(\mathfrak{A},B,c)$ be a class of $\{f_k (x)\}$ with $a\in \mathfrak{A},0<\delta<\sum\nolimits_{i,j,k=1}{b_{ij}^{(k)}}<B<\infty,c_{i,j,l}^{(k)}<c<\infty$. The following asymptotic formula for $t\to\infty,\lambda\to0$ holds true uniformly for all $\{{f_k}\}\in\mathrm K$
$$1-\mathbf P\{\mu_{ij}(t)=0,j=1,\dots,n\bigl|\mu_i>0\}\sim\mu_i k(t,\lambda,0),$$
where $k(t,\lambda ,x)$ is given by (7), $\mu _i \sum\nolimits_{j=1}^n{\mu _{ij}(t)}$. The probability distributions
$$S_k^{(t)}(y_1,\dots y_n)=\mathbf P\left\{\frac{\mu_{kj}(t)}{\mathbf M\{\mu _{kj}|\mu_k<0\}}<y_j,j=1,\dots,n,y_j ,j=1,\dots,n\bigr|\mu _k > 0\right\}$$
converge to an exponential distribution as $t \to \infty ,\lambda \to 0$, uniformly for all $\{f_k\}\in\mathrm K$.