On random walks and levels of the form \(n^ \alpha\)
Let \(X_i\), \(i \geq 1\), be i.i.d. random variables with finite mean \(\mu\) and let \(S_n = \sum^n_{k =1} X_i\), \(n \geq 1\), be the associated random walk. The number \(N\) of indices \(n\) with \(|S_n - n \mu|> \varepsilon n^\alpha\) is for \(\alpha > 1/2\) and \(\varepsilon > 0\) an a.s. finite random variable. This is also true for the number \(N_+\) of \(n\) such that \(S_n > n(\mu + \varepsilon)\) and the first index \(T\) for which this last inequality holds. The authors obtain results on the expected value of \(N\) for normally distributed \(X_i\). Bounds for the first two moments of \(N_+\) and properties of the distribution of \(T\) are stated under the assumption that the \(X_i\) have a second moment.
scientific article; zbMATH DE number 894674