Let $W$ be a Wiener process. Prove that the upper limit $$\limsup_{T\to\infty}\frac{\int_0^TW^2(s)\,ds}{T^2\ln\ln T}$$ is non-random (with probability 1) and calculate it. I know that I have to use Strassen's law of iterated logarithm, so that this limit is $$\sup_{f\in K}\int_0^1f^2(s)ds$$ where $K$ is the class of absolutely continuous functions such that $\int_0^1(f'(s))^2ds<1$ and $f(0)=0$, I can find some bounds for this supremum using Cauchy-Schwarz inequality, but I can't find exact value.
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zhoraster
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1In other words, you have to calculate the norm of the operator $h \mapsto \int_0^\cdot h(s) ds$ in $L^2[0,1]$. See here and here. – zhoraster Apr 10 '20 at 13:56