Past exam of the martingales and financial mathematics course of the University of Oxford / 2009 / A / 3 / a / ii / Solution

Each increment has the same law as its negative, and their independence makes the joint law of $(X_1,X_2,\dots )$ invariant under simultaneous sign reversal. Under this transformation, the simple symmetric random walk $S_n$ becomes $-S_n$, and the event $B_{-}$ becomes $B_{+}$. ThusThis uses symmetry of the entire increment sequence, rather than just the one-dimensional distributions of the positions.