Level Crossing Probabilities I: One-dimensional Random Walks and Symmetrization
نویسنده
چکیده
Abstract. We prove for an arbitrary one-dimensional random walk with independent increments that the probability of crossing a level at a given time n is O(n). 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 higherdimensional walks and some conjectures on directionally random walks in the sense of Mauldin, Monticino and v.Weizsäcker [4].
منابع مشابه
Level Crossing Probabilities Ii: Polygonal Recurrence of Multidimensional Random Walks
In part I (math.PR/0406392) we proved for an arbitrary onedimensional random walk with independent increments that the probability of crossing a level at a given time n is O(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 e...
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تاریخ انتشار 2004