Fixed points with finite mean of the smoothing transform in random environments

Abstract

At each time $n\in \mathbb{N}$, let ${\bar{Y}}^{(n)}=(y_1^{(n)},y_2^{(n)},\cdots )$ be a random sequence of non-negative numbers that are ultimately zero in a random environment $\xi =({\xi }_n{)}_{n\in \mathbb{N}}$ in time, which satisfies for each $n\in \mathbb{N}$ and a.e. $\xi ,E_{\xi }[{\sum }_{i\in {\mathbb{N}}_{+}}{y}_i^{(n)}(\xi )]=1.$ The existence and uniqueness of the non-negative fixed points of the associated smoothing transform in random environments is considered. These fixed points are solutions of the distributional equation for $a.e.\xi ,Z(\xi )\stackrel{d}{=}{\sum }_{i\in {\mathbb{N}}_{+}}{y}_i^{(0)}(\xi )Z_i(T\xi ),$ where when given the environment $\xi$, $Z_i(T\xi )(i\in {\mathbb{N}}_{+})$ are $i.i.d.$ non-negative random variables, and distributed the same as $Z(\xi )$. As an application, the martingale convergence of the branching random walk in random environments is given as well. The classical results by Biggins (1977) has been extended to the random environment situation.