So the Levy's modulus of continuity theorem says that almost surely,
$\limsup_{h \to 0} \sup_{0 \leq t \leq 1-h} \frac{B(t+h)-B(t)}{\sqrt{2h \log\frac{1}{h}}}=1.$
while Khinchtine's Law of Iterated Logarithm says that almost surely,
$\limsup_{t\to \infty} \frac{B(t)}{\sqrt{2 \frac{1}{t}\log \log t}}=1.$
Using the fact that $tB(\frac{1}{t})$ is also a Brownian motion and letting $h=\frac{1}{t}$, we get:
$\limsup_{h \to 0} \frac{B(h)}{\sqrt{2h\log \log\frac{1}{h}}}=1.$
Now I'm finding it difficult to understand the difference between these two statements. If we put $t=0$ in the first statement, we get $\limsup_{h \to 0}\frac{B(h)}{\sqrt{2h \log\frac{1}{h}}}=1$.
I thought Levy's modulus of continuity was optimal for the behaviour of Brownian motion near $0$.
Is the only difference that Levy's modulus controls the supremum?
Best Answer
About the small increments $B(t+h)-B(t)$ of BM(Brownian motion), there are two classes of results:
(i) LIL(Law of Iterated Logarithm) for fixed start point $t$;
(ii) Levy's modulus of continuity for moving start points.
(i) For fixed start point--LIL :
\begin{gather*} \varlimsup_{\delta\downarrow0}\frac{|B(t+\delta)-B(t)|}{ \sqrt{2\delta \log\log\delta^{-1}}}=1.\quad \text{a.s.}\quad \forall t>0. \\ \varlimsup_{\delta\downarrow0} \frac{\sup_{0< h\le \delta} |B(t+h)-B(t)|}{\sqrt{2\delta\log\log \delta^{-1}}}=1.\quad \text{a.s.} \quad \forall t>0. \end{gather*}
(ii) For moving start points--Levy's modulus: \begin{gather*} \varlimsup_{\delta\downarrow0}\frac{\sup_{0< t\le 1}|B(t+\delta)-B(t)|}{ \sqrt{2\delta \log\delta^{-1}}}=1.\quad \text{a.s.}\\ \varlimsup_{\delta\downarrow0}\frac{\sup_{0< t\le 1} \sup_{0< h\le \delta} |B(t+h)-B(t)|}{\sqrt{2\delta\log\delta^{-1}}}=1.\quad \text{a.s.}\\ \varlimsup_{\delta\downarrow0}\frac{\sup_{0<s, t\le 1, |s-t|<\delta} |B(t)-B(s)|}{\sqrt{2\delta\log\delta^{-1}}}=1.\quad \text{a.s.} \end{gather*}
Remark: in the book M. Csörgo and P.Révész, Strong Approximations in Probability and Statics, Academic Press, 1981. $\S1.2$ p.29--., there are more results about "how big are the increments of a Wiener process".