Let be a critical, discrete-time, one-type Markov branching process in a random environment , i.i.d copies of , having probability measures on as its values. Define , , , let be the random walk given by . Suppose is a.s. finite and define when , and otherwise, where the are the strict descending ladder epochs of . Set , , where . Consider two sets of assumptions:
(A) There exits a , , such that , and, for some and some integer , and .
(B) The distribution of belongs without centering to the domain of attraction of some stable law which is not one-sided and has index , , and, for some and some integer , .
First, assume (A) or (B). Then, for some finite positive real number ,
and, as a corollary,
with slowly varying at infinity. For any sequence of integers such that and , as , the conditional distribution of converges weakly (Skorokhod topology) to the law of a process with a.s. constant paths, a.s.. The distribution of converges weakly to a probability measure. Assuming (B), there exists a sequence slowly varying at infinity, such that the conditional distribution of converges weakly to the law of the meander of a strictly stable process with index, and, as a corollary, the corresponding result for in place of holds.