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Consider the invariance principle for a random walk with random environment(denoted by $\mu$) in time on $\bfR$ in a weak quenched sense. We show that asequence of the random probability measures on $\bfR$ generated by a boundedLipschitz functional $f$ and $\mu$ will converge in distribution to anotherrandom probability measures, which is related to $f$ and two independentBrownian motions. The upper bound of the convergence rate has been obtained. Wealso explain that in general, this convergence can not be strengthened to thealmost surely sense.

Weak Quenched Invariance Principle for Random Walk with Random Environment in Time