Let \(\{ Z_n: n=0,1, \cdots\}\) be a collection of mutually independent random variables with \(Z_n\) distributed as a Gaussian with mean zero and variance \(\sigma_n^2\) for \(\sigma_n \in \mathbb{R}\). Define \(X_n\) by

\begin{equation*}

X_n=\sum_{k=0}^n Z_k^2\;.

\end{equation*}

- Find a stochastic process \(Y_n\) so that \(X_n-Y_n\) is a Martingale with respect to the filtration \(\mathcal{F}_n=\sigma(Z_0,\cdots,Z_n)\).
- Find a second process \(\tilde Y_n\) so that \(X_n-Y_n\) is again a Martingale with respect to the filtration \(\mathcal{F}_n\) but\(Y_n \neq \tilde Y_n\) almost surely.