Conditional $L^1$-Convergence for the Martingale of a Critical Branching Process in Random Environment

For a critical branching process $(Z_n)$ in a random environment $(\xi _n)$, a sufficient condition is given for the corresponding martingale ${Z_n}/{e^{S_n}}$ to converge in $L^1$ or to degenerate under $\mathbb P^+$, the probability under which the associated random walk is conditioned to stay nonnegative.