On a new law of the iterated logarithm of Erdős and Révész (Q1320487)

Let \(\{W(t): t\geq 0\}\) be a standard Wiener process and define the processes \[ \begin{aligned} X(t) & =\sup\bigl\{s: 0\leq s\leq t,\;W(s)\geq (2s\log\log s)^{1/2}\bigr\},\quad t\geq 0,\\ X_ \varepsilon(t) & =\sup\bigl\{s: 0\leq s\leq t,\;W(s)\geq (2(1-\varepsilon)s\log\log s)^{1/2}\bigr\},\quad t\geq 0,\;0\leq \varepsilon< 1,\\ X^{(p)}_ \delta (t) & =\sup\bigl\{s: 0\leq s\leq t,\;W(s)\geq s^{1/2} \alpha(\delta,p,s)\bigr\},\quad t\geq 0,\end{aligned} \] where \[ \alpha(\delta,p,s)= \left(2\left(\log_ 2 s+{\textstyle{{3\over 2}}}\log_ 3 s+ \sum^ p_{j=4} \log_ j s- \delta\log_ p s\right)\right)^{1/2},\quad \delta\geq 0,\;p=3,4,\dots\;. \] Answering some challenging questions of \textit{P. Erdős} and \textit{P. Révész} [Acta Math. Hung. 55, No. 1/2, 125-131 (1990; Zbl 0711.60025)], the author finds the exact lower bounds for the processes \(X(t)\), \(X_ \varepsilon(t)\) and \(X^{(p)}_ \delta(t)\).