这是一份umanitoba曼尼托巴大学STAT 7060作业代写的成功案
Let $\left{X_{n}, n \geq 1\right}$ be iid with common continuous distribution function $F(x)$. The continuity of $F$ implies
$$
P\left[X_{i}=X_{j}\right]=0
$$
so that if we define
$$
\text { [Ties }]=\bigcup_{t \neq \jmath}\left[X_{i}=X_{j}\right] \text {, }
$$
then
$$
P[\text { Ties }]=0 .
$$
Call $X_{n}$ a record of the sequence if
$$
X_{n}>\bigvee_{t=1}^{n-1} X_{i}
$$
.
STAT 7060 COURSE NOTES ：
Also,
$$
\mathcal{C}{T{s}} \supset \mathcal{B}{\alpha}, \quad \forall \alpha \in T{s}
$$
(we can take $K={\alpha}$ ) and hence
$$
\sigma\left(\mathcal{C}{T{s}}\right) \supset \mathcal{B}{\alpha}, \quad \forall \alpha \in T{s}
$$
It follows that
$$
\sigma\left(\mathcal{C}{T{s}}\right) \supset \bigcup_{\alpha \in T_{s}} \mathcal{B}{\alpha} $$ and hence $$ \sigma\left(\mathcal{C}{T_{s}}\right) \supset \sigma\left(\bigcup_{\alpha \in T_{s}} \mathcal{B}{\alpha}\right)=: \bigvee{\alpha \in T_{s}} \mathcal{B}_{\alpha}
$$