Let \(0\le X_n\le 1\) be adapted to \(\mathcal F_n\). Let \(\alpha ,\beta >0\) with \(\alpha +\beta =1\) and suppose \[ \mathbb {P}(X_{n+1}=\alpha +\beta X_n\mid \mathcal F_n)=X_n \] \[ \mathbb {P}(X_{n+1}=\beta X_n\mid \mathcal F_n)=1-X_n. \]
(a) Show that \(\mathbb {P}\{\lim _{n\to \infty }X_n=0\text { or }1\}=1\).
(b) Show that if \(X_0=\theta \), then \(\mathbb {P}\{\lim _{n\to \infty }X_n=1\}=\theta \).