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

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