We prove that the centered three-dimensional Wiener sausage can be strongly approximated by a one-dimensional Brownian motion
running at a suitable time clock. The strong approximation gives all possible laws of iterated logarithm as well as the convergence
in law in terms of process for the normalized Wiener sausage. The proof relies on Le Gall șs fine L2-norm estimates between the Wiener sausage and the Brownian intersection local times.
A certain class of stochastic summability methods of mantissa type is introduced and its connection to almost sure limit theorems
is discussed. The summability methods serve as suitable weights in almost sure limit theory, covering all relevant known examples
for, e.g., normalized sums or maxima of i.i.d. random variables. In the context of semistable domains of attraction the methods
lead to previously unknown versions of semistable almost sure limit theorems.