 Positive random-environment path-weight martingale

Let $\xi$ be a random walk, independent of centered, independent time-space signs $h$, and let $0<\epsilon <1$. Averaging over the walk gives a positive martingale in the environment filtration: each new time layer has conditional mean-one weight. Hence $\mathbb{E}{Z}_t=1$ and the martingale convergence theorem gives a finite nonnegative almost-sure limit. Preserving its mean requires an additional condition such as bounded second moments; positivity of finite-time weights alone does not suffice.