Math 6010 • Linear Models

Let \(0\leq X_n\leq 1\) be adapted to \(\mathcal{F}_n\). Let \(\alpha,\beta>0\) with \(\alpha+\beta=1\) and suppose
\[
P(X_{n+1}=\alpha+\beta X_n\mid \mathcal{F}_n)=X_n
\]
\[
P(X_{n+1}=\beta X_n\mid \mathcal{F}_n)=1-X_n.
\]
\begin{parts}
\item Show that \(P\{\lim_{n\to\infty}X_n=0\text{ or }1\}=1\).
\item Show that if \(X_0=\theta\), then \(P\{\lim_{n\to\infty}X_n=1\}=\theta\).
\end{parts}