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Mar 2, 2020 at 19:58 comment added Mateusz Kwaśnicki In the displayed equation, the third equality is wrong: we have $\mathbb{E}_x[T_y \circ \theta_{T_z} | \mathscr{F}_{T_z}] = \mathbb{E}_z[T_y]$ a.s., but $T_y \circ \theta_{T_z} = T_y - T_z$ only on the event $\{T_y > T_z\}$. Also, there are extremely few functions that satisfy $d(x,y) = d(x,z) + d(z,y)$ for every triple $x, y, z$...
Mar 2, 2020 at 16:33 history answered user1118 CC BY-SA 4.0