to random offspring distributions which vary from one generation to the other. The present article complements results established in [VD]{}-[@VD2008]  where critical branching processes in random environment were investigated under the quenched approach. To give a formal description of the model under consideration we shall spend some time in this section introducing notation before proceeding to the main results in the next section.

Let $\Delta $ be the space of probability measures on $\mathbb{N}_{0}:=\{0,1,2,...\}$. Equipped with the metric of total variation $\Delta $ becomes a Polish space. Let $\mathbf{K}$ be a random variable taking values in $\Delta $. An infinite sequence $\mathbf{\bar{K}}=(\mathbf{K}_{0},\mathbf{K}_{1},\ldots )$ of i.i.d. copies of $\mathbf{K}$ is said to form a *random environment*. A sequence of $\mathbb{N}_{0}$-valued random variables $Z_{0},Z_{1},\ldots $ is called a *branching process in the random environment* $\mathbf{\bar{K}}$, if $Z_{0}$ is independent of $\mathbf{\bar{K}}$ and given $\mathbf{\bar{K}}$ the process $Z=(Z_{0},Z_{1},\ldots )$ is a Markov chain with $$\mathcal{L}\left( Z_{n+1}\;|\;Z_{n}=z(n),\,\mathbf{\bar{K}}=(\mathbf{k}_{0},\mathbf{k}_{1},\ldots )\right) \ =\ \mathcal{L}\left( \xi _{n1}+\cdots +\xi
_{nz(n)}\right)  \label{transition}$$for every $n,\,z(n)\in \mathbb{N}_{0}$ and $\mathbf{k}_{0},\mathbf{k}_{1},\ldots \in \Delta $, where $\xi _{n1},\xi _{n2},\ldots $ are i.i.d. random variables with distribution $\mathbf{k}_{n}$. Setting$$f_{n}(s):=\sum_{j=0}^{\infty }\mathbf{k}_{n}(\{j\})s^{j}$$one can rewrite (\[transition\]) as$$\mathbf{E}\left[ s^{Z_{n+1}}|\;Z_{n}=z(n),\,\mathbf{\bar{K}}=(\mathbf{k}_{0},\mathbf{k}_{1},\ldots )\right] =\left( f_{n}\left( s\right) \right)
^{z(n)},\,n\geq 0.$$

Let $\Omega =\left\{ \omega 