Level crossing probabilities I: One-dimensional random walks and symmetrization

We prove for an arbitrary random walk in R¹ with independent increments that the probability of crossing a level at a given time n is O(n−1/2). Moment or symmetry assumptions are not necessary. In removing symmetry the (sharp) inequality P(|X+Y|?1)<2P(|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 reinforced random walks in the sense of Mauldin, Monticino and von Weizsäcker R.D. Mauldin, M. Monticino, H. von Weizsäcker, Directionally reinforced random walks, Adv. Math. 117 (1996) 239-252. 5]].