Accuracy of approximation in the Poisson theorem in terms of the -distance.
We consider a multidimensional random walk in a product random environment with bounded steps, transience in some spatial direction, and high enough moments on the regeneration time. We prove an invariance principle, or functional central limit theorem, under almost every environment for the diffusively scaled centered walk. The main point behind the invariance principle is that the quenched mean of the walk behaves subdiffusively.
We prove an almost sure functional limit theorem for the product of partial sums of i.i.d. positive random variables with finite second moment.