#### Vol. 3, No. 4, 2021

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Central limit theorem for a critical multitype branching process in random environments

### Emile Le Page, Marc Peigné and Da Cam Pham

Vol. 3 (2021), No. 4, 801–842
##### Abstract

Let ${\left({Z}_{n}\right)}_{n\ge 0}$ with ${Z}_{n}={\left({Z}_{n}\left(i,j\right)\right)}_{1\le i,j\le p}$ be a $p$ multitype critical branching process in random environment, and let ${M}_{n}$ be the expectation of ${Z}_{n}$ given a fixed environment. We prove theorems on convergence in distribution of sequences of branching processes $\left\{\left({Z}_{n}∕|{M}_{n}|\right)\mid |{Z}_{n}|>0\right\}$ and $\left\{\left(ln{Z}_{n}∕\sqrt{n}\right)\mid |{Z}_{n}|>0\right\}$. These theorems extend similar results for single-type critical branching process in random environments.

 Emile Le Page passed away on May 2021, after a lengthy battle with cancer. We have lost a generous colleague and a friend. — Marc Peigné and Da Cam Pham
##### Keywords
multitype branching process, random environment, central limit theorem
##### Mathematical Subject Classification
Primary: 60J80, 60K37
Secondary: 60F17