Functional limit theorems for strongly subcritical branching processes in random environment

For a strongly subcritical branching process (Zn)n[greater-or-equal, slanted]0 in random environment the non-extinction probability at generation n decays at the same exponential rate as the expected generation size and given non-extinction at n the conditional distribution of Zn has a weak limit. Here we prove conditional functional limit theorems for the generation size process (Zk)0[less-than-or-equals, slant]k[less-than-or-equals, slant]n as well as for the random environment. We show that given the population survives up to generation n the environmental sequence still evolves in an i.i.d. fashion and that the conditioned generation size process converges in distribution to a positive recurrent Markov chain.