Let (left(B_{t} ight)_{t geqslant 0}) be a (mathrm{BM}^{1}). Apply Doob's maximal inequality (A.13) to the exponential martingale

Let \(\left(B_{t}\right)_{t \geqslant 0}\) be a \(\mathrm{BM}^{1}\). Apply Doob's maximal inequality (A.13) to the exponential martingale \(M_{t}^{\xi}:=\exp \left(\xi B_{t}-\frac{1}{2} \xi^{2} t\right)\) to show that for all \(x, \xi>0\)\[\mathbb{P}\left(\sup _{s \leqslant t}\left(B_{s}-\frac{1}{2} \xi s\right)>x\right) \leqslant e^{-x \xi}\]

(This inequality can be used for the upper bound in the proof of Theorem 12.1, avoiding the combination of (12.2) and the upper bound in (12.1).)

Data From 12.1 Theorem 

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12.1 Theorem (LIL. Khintchine 1933). Let (B(t))o be a BM. Then Plim B(t) 2t log log t -1)=1. (12.3) Since (-B(t))20 and (tB(t))so are again Brownian motions, we can immediately get versions of the Khintchine LIL for both the limit inferior and t 0.