Let (X_{0}) be uniformly distributed over ([0, T], X_{1}) be uniformly distributed over (left[0, X_{0} ight]), and,

Let \(X_{0}\) be uniformly distributed over \([0, T], X_{1}\) be uniformly distributed over \(\left[0, X_{0}\right]\), and, generally, \(X_{i+1}\) be uniformly distributed over \(\left[0, X_{i}\right], i=0,1, \ldots\).

Verify: The sequence \(\left\{X_{0}, X_{1}, \ldots\right\}\) is a supermartingale with \(E\left(X_{k}\right)=\frac{T}{2^{k+1}} ; k=0,1, \ldots\)