Suppose I have a continuous function $f:[0,1]\to\mathbb{R}$, and I wish to measure somehow how similar it is, in some sense, to a Brownian motion $\{B(t)\mid t\in[0,1]\}$ (with $B(0)=0$).
I was wondering whether there are off-shelf measures for such a similarity, perhaps something like the Berry–Esseen theorem... if not, I was coming up with an idea, and I wonder what do you think about it:
- Let $T = \{(t,B(t))\mid t\in[0,1]\}$ be the (random) trajectory of a Brownian motion at times $[0,1]$.
- For a continuous function $f:[0,1]\to\mathbb{R}$, let $f^{(\varepsilon)}=\{(t,f(t)+\delta t)\mid t\in[0,1],\ \delta\in[-\varepsilon,\varepsilon]\}$ be the "expanded" graph of $f$.
- Define $\rho^{(\varepsilon)}:C([0,1])\to[0,1]$ to be the "$\varepsilon$-typicalness" measure of continuous functions on $[0,1]$ as follows: \begin{equation*} \rho^{(\varepsilon)}(f) = \mathbb{P}\left(T\subseteq f^{(\varepsilon)}\right). \end{equation*}
I am not entirely sure what happens near $0$. Do you think that the $\varepsilon$-typicalness of every function $f$, for every $\varepsilon$, will be $0$ as it is defined now? If so, can you think of a minor correction?