Here on [wikipedia][1] is claimed that the process $X_t:=\sup_{s \in (0,t)} B_s-B_t$ is distributed like $\vert B_t \rvert$ where $B_t$ is standard Brownian motion.

On the other hand, it is claimed [here in Corollary $6.21$][2] that $\sup_{s \in (0,t)} B_s$ is distributed like $\vert B_t \rvert.$


So how is it possible that $\sup_{s \in (0,t)} B_s-B_t$ is distributed like $\sup_{s \in (0,t)} B_s.$ There seems to be something wrong with probability.

If you have any further questions, please let me know.


  [1]: https://en.wikipedia.org/wiki/Arcsine_laws_(Wiener_process)#Equivalence_of_the_second_and_third_laws
  [2]: http://www.statslab.cam.ac.uk/~ps422/mynotes.pdf