Controlled branching processes

We define the controlled branching process $\mu(t)$ as a process in which the number of particles $\mu(t+1)$ in the $(t+1)$-th generation equals the sum:
$$ \xi_1(t)+\dots+\xi_{\varphi(\mu(t))}(t), $$
where $\xi_i(t)$, $i=1,2,\dots$, $t=1,2,\dots$, are independent identically distributed integer-valued random variables independent of $\mu(t)$, $\varphi(n)$ is a non-negative integer-valued function.
We investigate asymptotic properties of such processes when (1) $\varphi(n)\sim\alpha n$ or (2) $\varphi(n)\sim cn^\beta$ as $n\to\infty$. Let $A=\mathbf M\xi_i(t)$ in case (1) and $\mathbf P\{\xi_i(t)> x\}\sim cx^{-\alpha}$ as $x\to\infty$ in case (2). We prove that the process is subcritical if $\alpha A<1$ in case (1) and if $\beta<\alpha$ in case (2), and is supercritical if $\alpha A>1$ in case (1) and if $\beta>\alpha$ in case (2).