  We prove for an arbitrary one-dimensional random walk with independent
increments that the probability of crossing a level at a given time n has the
order of square root of n. Moment or symmetry assumptions are not necessary. In
removing symmetry the (sharp) inequality P(|X+Y| <= 1) < 2 P(|X-Y| <= 1) for
independent identically distributed X,Y is used. In part II we shall discuss
the connection of this result to 'polygonal recurrence' of higher-dimensional
walks and some conjectures on directionally random walks in the sense of
Mauldin, Monticino and v.Weizsaecker [5].
