On the large values of the Wiener process (Q1096246)

Let \(\{W(t),t\geq 0\}\) be a standard Wiener process and define \(M^+(t)=\max \{W(u):\) \(u\leq t\}\), \(M^-(t)=\max \{-W(u):\) \(u\leq t\}\) and \(Z(t)=\max \{u\leq t:\) \(W(u)=0\}\). The authors investigate the a.s. asymptotic behaviour of Z(t) and \(M^-(t)\) under the condition that \(M^+(t)\) gets very large, i.e. as large as indicated by the law of iterated logarithm. By using Skorohod's embedding scheme, the similar conclusions are given.