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Suppose I have two Wiener processes, which are independent - call them $B(t)$ and $W(t)$. I think it should be true that

$$\liminf_{T \rightarrow \infty} \frac{\ln\ln T}{T^2}\left|\sup_{0 \leq x \leq T} \int_{0}^{x} W(t)B(t)dt \right|>0.$$

But I am not sure how to prove it... any ideas will be very much appreciated.

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