Recursive martingale

Solution 1:

I think you are skipping some non-trivial steps here. First, it is not readily apparent that $X_n$ is integrable. From the definition of $X_1$, we see that its minimum value is zero (attained at $x_0=0$ with probability one) and its maximum value is one (attained at $x_0=1$ with probability one). Then, assuming $0\leqslant X_n\leqslant 1$, we have by a similar argument that $0\leqslant X_n\leqslant1$, so by induction $0\leqslant X_n\leqslant 1$ for all $n$. It follows then that \begin{align} \mathbb E[|X_n|] &= \mathbb E[X_n]\\ &= (1/2 + X_n/2)X_n + X_n/2(1-X_n)\\ &\leqslant (1/2+1/2)\cdot 1 + 1/2\cdot 1\\ &= 5/2\\&<\infty. \end{align}

Your verification of the martingale property is correct.

As for convergence, I believe my argument above suffices for Doob's martingale convergence theorem. (Technically the condition is $\sup_n \mathbb E[X_n^-]<\infty$ but $\mathbb P(X_n\geqslant0)$ here so we need not consider the negative part of $X_n$.)