A Chung type law of the iterated logarithm for subsequences of a Wiener process (Q1904539)

This paper studies conditions for \(\lim \inf_{n \to \infty} \alpha_n^{-1} \max_{i \leq n} W(t_i) = \gamma\) where \(W(t)\) is a standard Wiener process, \(\{t_n\}\) a real increasing sequence converging to \(\infty\) and \(\{\alpha_n\}\), \(\gamma\) constants defined in terms of \(\{t_n\}\). The classical case is Chung's iterated logarithm law where \(t_n = n\), \(\alpha_n = (n/ \log \log n)^{1/2}\), \(\gamma = \pi/ \sqrt 8\), and the author (1992) proved that if \[ t_n - t_{n - 1} = o(t_n/ \log \log t_n), \tag{*} \] one can take \(\alpha_n = (t_n/ \log \log t_n)^{1/2}\), \(\gamma = \pi/ \sqrt 8\). This paper gives results for sequences \(\{t_n\}\) not satisfying (*), where \(\alpha_n\) and \(\gamma\) may be different. The results imply in particular that one cannot replace \(o (\cdot)\) in (*) by \(O (\cdot)\).