= Tail event for a vanishing positive path-weight limit
Delete the first $m$ environment factors from the path-weight expectation and call the result $Z_T^{(m)}$. Pointwise, $(1-\varepsilon)^mZ_T^{(m)}\le Z_T\le(1+\varepsilon)^mZ_T^{(m)}$. Thus $\{\liminf_TZ_T=0\}$ equals $\{\liminf_TZ_T^{(m)}=0\}$ and belongs to every future-time environment sigma-algebra. The <Kolmogorov zero-one law> makes its probability zero or one. If the nonnegative <martingale> limit has <mean> one, the probability cannot be one, so its limit is strictly positive almost surely. The pointwise comparison establishes a genuine <tail event>, rather than merely invariance outside an unspecified exceptional set.
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