  In this work we study the set of eventually always hitting points in
shrinking target systems. These are points whose long orbit segments eventually
hit the corresponding shrinking targets for all future times. We focus our
attention on systems where translates of targets exhibit near perfect mutual
independence, such as Bernoulli schemes and the Gauss map. For such systems, we
present tight conditions on the shrinking rate of the targets so that the set
of eventually always hitting points is a null set (or co-null set
respectively).
