An unfair fair game. Define random variables recursively by Y0 D 1 and for n 1;
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An unfair fair game. Define random variables recursively by Y0 D 1 and for n 1; Yn is chosen uniformly on .0; Yn1/. If we let U1; U2; : : : be uniform on
.0; 1/, then we can write this sequence as Yn D UnUn1 U0. (a)Use Example 5.5 to conclude that Mn D 2nYn is a martingale.
(b) Use the fact that log Yn D log U1 C ClogUn to show that .1=n/ logXn !1.
(c) Use
(b) to concludeMn ! 0, i.e., in this “fair” game our fortune always converges to 0 as time tends to1.
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