Show that (mathrm{BM}^{1}) is almost surely not (1 / 2)-Hlder continuous: a) For all (Z sim mathrm{N}(0,1))

Show that \(\mathrm{BM}^{1}\) is almost surely not \(1 / 2\)-Hölder continuous:

a) For all \(Z \sim \mathrm{N}(0,1)\) and \(x>0\) we have

\[\frac{1}{\sqrt{2 \pi}} \frac{x e^{-x^{2} / 2}}{x^{2}+1}<\mathbb{P}(Z>x)<\frac{1}{\sqrt{2 \pi}} \frac{e^{-x^{2} / 2}}{x} .\]

b) Define \(A_{k, n}:=\left\{\left|B\left((k+1) 2^{-n}ight)-B\left(k 2^{-n}ight)ight|>c \sqrt{n 2^{-n}}ight\}\) and show that for each \(c<\sqrt{2 \log 2}\) one has \(\mathbb{P}\left(\overline{\lim }_{n ightarrow \infty} \bigcup_{k=0}^{2^{n}-1} A_{k, n}ight)=1\). For this, use the lower bound in a) and consider the complement of the set \(\varlimsup_{n ightarrow \infty} \bigcup_{k=0}^{2^{n}-1} A_{k, n}\).

c) Conclude from the previous part, that \(\mathrm{BM}^{1}\) is a.s. not \(1 / 2\)-Hölder continuous.