I'm supposed to show that there exists a $c > 0 $ such that, for each $P\left(|B_t| \leq \varepsilon, \forall t \in [0, 1]\right) \leq e^{-\frac{c}{{\varepsilon}^2}}$. We're given the hint: $B_{k\varepsilon^2} - B_{(k-1)\varepsilon^2} \quad \text{for } k = 1, \ldots, \left\lfloor \frac{1}{\varepsilon^2} \right\rfloor$ however frankly I'm just completely lost. I understand the hint will give me a sum of independent intervals that sums up to $B_t$, however, I have no idea how I'm supposed to use this. In my mind, I need to use the reflection principle here, but I'm not sure how that's supposewd to interact with the hint I'm given, and I've looked at just reexpressing the condition by looking at the squares of $B_t$ and $\varepsilon$ but it doesn't seem I can apply the reflection principle then, so I'm struggling a bit here. Any help would be appreciated.
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Look up the proof of Schilder’s theorem; this would be part of it – Andrew Nov 30 '23 at 22:15
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I'm having trouble finding an explicit proof, from what I've found most of it is just statements of the theorem and examples of its usage – trgjtk Nov 30 '23 at 22:30
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okay nevermind, i got it by looking at the probability of each interval being within $2\epsilon$ thanks for the input – trgjtk Dec 01 '23 at 04:29