Asymptotic behavior of the extinction probabilitiesfor stopped branching processes

The initial multitype Galton–Watson branching process
$$ \mu(t)=(\mu_1(t),\mu_2(t),\dots,\mu_m(t)), \qquad t=0,1,2,\dots, $$
generates a stopped branching process $\xi(t)$, if the evolution of $\mu(t)$ is ‘`frozen’ when it hits a finite set $S$. It is assumed that the initial branching process $\mu(t)$ is subcritical and indecomposable. We prove that the extinction probability
$$ q_r^n=\lim_{t\to\infty}\mathsf{P}\{\xi(t)=r\mid\xi(0)=n\} $$
is asymptotically approaching, for any $r=(r_1,r_2\ldots r_m)\in S$, $n=(n_1\ldots n_m)\notin S$ for $\overline n=n_1+\dots+n_m\to\infty$, $n_i/\overline n\to a_i$, a function which is periodic in $\log_{1/R}\overline n$ with period 1. Here $R < 1$ is the Perron root of the mean matrix of the initial subcritical branching process $\mu(t)=(\mu_1(t),\mu_2(t)\ldots \mu_m(t))$ with elements $A_{ij}=\mathsf{E}\{\mu_j(1)\mid\mu(0)=e(i)\}$, and $e(i)=(\delta_{i1},\delta_{i2},\dots,\delta_{im})$, where $\delta_{ij}$ is the Kronecker symbol.