This is an addendum to Problem where a stick of length 1 was repeatedly randomly broken in the sense that the remaining piece each time was U(0, Y )-distributed, where Y was the (random) previous length. Let Yn denote the remaining piece after the stick has been broken n times, and set Fn = σ{Yk, k ≤ n}.
(a) Compute E(Yn|Fn-1).
(b) Adjust Yn in order for (a suitable) {(Xn, Fn), n ≥ 0} to become a martingale.
(c) Does Xn converge almost surely? In L1?