Limit theorems for critical branching processes in a finite state space Markovian environment

Let $(Z_n{)}_{n\ge 0}$ be a critical branching process in a random environment defined by a Markov chain $(X_n{)}_{n\ge 0}$ with values in a finite state space $\mathbb{X}$. Let $S_n={\sum }_{k=1}^n\ln f_{X_k}^{\mathrm{\prime }}(1)$ be the Markov walk associated to $(X_n{)}_{n\ge 0}$, where $f_i$ is the offspring generating function when the environment is $i\in \mathbb{X}$. Conditioned on the event $\{Z_n>0\}$, we show the non degeneracy of limit law of the normalized number of particles $Z_n\mathrm{/}{e}^{S_n}$ and determine the limit of the law of $\frac{S_n}{\sqrt{n}}$ jointly with $X_n$. Based on these results we establish a Yaglom-type theorem which specifies the limit of the joint law of $\log Z_n$ and $X_n$ given $Z_n>0$.