Math 6010 • Linear Models

Let \(X_1,X_2,\ldots\) be independent random variables with
\[
  X_n=
  \begin{cases}
    1, & \text{with probability }1/(2n),\\
    0, & \text{with probability }1-1/n,\\
    -1, & \text{with probability }1/(2n).
  \end{cases}
\]
Let \(Y_1=X_1\) and for \(n\geq2\) define
\[
  Y_n=
  \begin{cases}
    X_n, & \text{if }Y_{n-1}=0,\\
    nY_{n-1}|X_n|, & \text{if }Y_{n-1}\ne0.
  \end{cases}
\]
Show that \(Y_n\) is a martingale with respect to
\(\mathcal{F}_n=\sigma(Y_1,\ldots,Y_n)\). Show that \(Y_n\) does not converge
almost surely. Why does the martingale convergence theorem not apply?