A Branching Process in Random Environment Starting with Large Number of Particles

Let $Z^{(k)}=\{ Z_i^{(k)},\, i=0,1,\dots\}$, $k=1,2,\dots$, be a sequence of critical branching processes in random environment, which differ from one another only in the population size $k$ of the initial generation. Suppose that the variance $\sigma^2$ of the step of the associated random walk is finite and positive. We fix $x\in (0,+\infty)$ and define $Z^{(n,x)}=Z^{(m_n(x))}$, where $m_1(x),m_2(x),\dots$ is a sequence of natural numbers, and $\ln m_n(x) \sim \sigma \sqrt{n}\, x$ as $n\to\infty$. We prove limit theorems on the extinction moment of the process $Z^{(n,x)}$, on the time-continuous normalized process constructed from $Z^{(n,x)}$, and on the normalized logarithm of the process $Z^{(n,x)}$.