Let $X_1,X_2,\dots$ be i.i.d. random variables and let $X_i^{(k)}=\min(X_i,k)$, $k=1,2,\dots$. Set $\mathcal{F}_n^{(k)}=\sigma\{X_1^{(k)},X_2^{(k)},\dots,X_n^{(k)}\}$ and $\mathcal{F}_n=\sigma\{X_1,X_2,\dots,X_n\}$.
(1) What's the relation between $\mathcal{F}_n^{(k)}$ and $\mathcal{F}_n$? Does $\mathcal{F}_n^{(k)}\to \mathcal{F}_n$ as $k\to\infty$ hold?
(2) If we let $M^{(k)}=\{\tau:\{\tau\le n\}\in\mathcal{F}_n^{(k)}\;\mbox{for all}\;n\}$ and $M=\{\tau:\{\tau\le n\}\in\mathcal{F}_n\;\mbox{for all}\;n\}$, what's the relation between $M^{(k)}$ and $M$? $M^{(k)}\to M$ as $k\to\infty$?
Any help or comment would be appreciated. Thanks.