Random walks have plenty of counter-intuitive properties when it comes to returning to the origin. - The average distance from the origin increases as time goe

Random walks have plenty of counter-intuitive properties when it comes to returning to the origin. - The average distance from the origin increases as time goes on. - But the average coordinate any given step ends up on is and remains the origin for the duration of the walk. More weirdly, though I don't know if this is true in two dimensions (ah, a sibling comment states it only holds in one dimension -- thanks!) - As time goes on, a single random walk will take longer and longer to return to the origin. - But a single random walk still returns to the origin infinitely often!