The law of iterated logarithm has the following implication for a standard Brownian motion $(W_t, t\geq 0)$, $$ \mathbb{P}\left(\limsup_{t\downarrow 0}\frac{W_t}{\sqrt{2t\ln\left(\ln\left(\frac{1}{t}\right)\right)}} = 1\right) = 1. $$ I wonder if there are results shedding light on the convergence rate of the above $\limsup$. To be more specific, what can I say about the following probability, $$ \mathbb{P}\left(W_t \leq \sqrt{2t\ln\left(\ln\left(\frac{1}{t}\right)\right)} + \varepsilon \right), $$ for a given $t$ and any $\varepsilon > 0$?
2026-03-26 08:07:28.1774512448
Convergence rate law of iterated logarithm for a Brownian motion
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