= Positive random-environment path-weight martingale
{title2=$Z_t=\mathbb E_\xi\prod_{j=1}^t(1+\varepsilon h(j,\xi_j))$}
Let $\xi$ be a <random walk>, <independent> of centered, <independent> time-space signs $h$, and let $0<\varepsilon<1$. Averaging over the walk gives a positive <martingale> in the environment <filtration>: each new time layer has conditional mean-one weight. Hence $\mathbb EZ_t=1$ 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.
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