Properties of multitype subcritical branching processes in random environment

Abstract: We study properties of a $p-$type subcritical branching process in random environment initiated at moment zero by a vector $\mathbf{z}=\left( z_{1},..,z_{p}\right) $\ of particles of different types. Assuming that the process belongs to the class of the so-called strongly subcritical processes we show that its survival probability to moment $n$\ behaves for large $n$\ as $C(\mathbf{z})\lambda ^{n}$\ where $\lambda $\ is the upper Lyapunov exponent for the product of mean matrices of the process and $C(\mathbf{z})$% \ is an explicitly given constant. We also demonstrate that the limiting conditional distribution of the number of particles given the survival of the process for a long time does not depend on the vector $\mathbf{z}$ of the number of particles initiated the process.