Level crossing probabilities. I: One-dimensional random walks and symmetrization (Q861049)

Let \(\{ X_i : i=1,2,\dots\}\) be iid random variables. Consider the one-dimensional random walk \(S_n = \sum_{i=1}^n X_i\). We say that the random walk crosses the level \(l\) at \(n\) if \(\text{sgn}(S_n - l) \neq \text{sgn}(S_{n-1} -l)\). The author's main result says that for any one-dimensional random walk \( S_n\) the probability of a crossing of level \(l\) at time \(n\) is \(O(n^{-1/2})\). [For part II see ibid., 680--698 (2007; Zbl 1108.60038).]