  In part I (math.PR/0406392) we proved for an arbitrary one-dimensional random
walk with independent increments that the probability of crossing a level at a
given time n is of the maximal order square root of n. In higher dimensions we
call a random walk 'polygonally recurrent' (resp. transient) if a.s. infinitely
many (resp. finitely many) of the straight lines between two consecutive sites
hit a given bounded set. The above estimate implies that three-dimensional
random walks with independent components are polygonally transient. Similarly a
directionally reinforced random walk on Z^3 in the sense of Mauldin, Monticino
and v.Weizsaecker [1] is transient. On the other hand we construct an example
of a transient but polygonally recurrent random walk with independent
components on Z^2.
