Mathematics – Probability
Scientific paper
2010-10-14
Mathematics
Probability
Scientific paper
Let $W(t), t\ge 0$ be 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 $z>0$, $$ \sup_{t\in I}{|W(t)|\over \sqrt t} \le z, $$ and we estimate their number of occurences. 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. Using Sakhanenko's invariance principe 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.
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