Math 6010 • Linear Models

Let \(X_1,X_2,\ldots\) be independent random variables such that
\[
  X_n=
  \begin{cases}
    1 & \text{with probability }(2n)^{-1},\\
    0 & \text{with probability }1-n^{-1},\\
   -1 & \text{with probability }(2n)^{-1}.
  \end{cases}
\]
Let \(Y_1:=X_1\) and for \(n\geq2\)
\[
  Y_n:=
  \begin{cases}
    X_n & \text{if }Y_{n-1}=0,\\
    nY_{n-1}|X_n| & \text{if }Y_{n-1}\neq0.
  \end{cases}
\]
Show that \(Y_n\) is a martingale with respect to
\(\mathcal{F}_n:=\sigma(Y_1,Y_2,\ldots,Y_n)\). Show that \(Y_n\) does not converge
almost surely. Does \(Y_n\) converge in any way? Why does the martingale
convergence theorem not apply?

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