Let W(t), t ≥ 0, be the standard Brownian motion. We study the size of the time intervals I which are admissible for the long range of slow increase, namely given a real z > 0, \(\sup _{t \in I} {{\left| {W(t)} \right|} \mathord{\left/ {\vphantom {{\left| {W(t)} \right|} {\sqrt t \leqslant z}}} \right. \kern-\nulldelimiterspace} {\sqrt t \leqslant z}}\). We obtain optimal results in terms of class test functions and, by means of the quantitative Borel-Cantelli lemma, a fine frequency result concerning their occurences. Some of these results extend to sums of independent random variables by means of Sakhanenko’s invariance principle. By also using this device to transfer the results to the Kubilius model, we derive applications to the prime number divisor function. We obtain refinements of some results recently proved by Ford and Tenenbaum in [4].
['Michel Weber']
Instants of small amplitude of brownian motion and application to the Kubilius model