To stimulate through theory and especially examples, an interest and appreciation of the power and elegance of martingales in analysis and probability. 

这是一份Bath巴斯大学MA40058作业代写的成功案

Let $\mathcal{K}$ be a vector subspace of $\mathcal{L}^{2}$ which is complete in that whenever $\left(V_{n}\right)$ is a sequence in $\mathcal{K}$ which has the Cauchy property that
$$
\sup {r, s \geq k}\left|V{r}-V_{s}\right| \rightarrow 0 \quad(k \rightarrow \infty)
$$
then there exists a $V$ in $\mathcal{K}$ such that
$$
\left|V_{n}-V\right| \rightarrow 0 \quad(n \rightarrow \infty)
$$
Then given $X$ in $\mathcal{L}^{2}$, there exists $Y$ in $\mathcal{K}$ such that
$$
|X-Y|=\Delta:=\inf {|X-W|: W \in \mathcal{K}}
$$
(ii)
$$
X-Y \perp Z, \quad \forall Z \in \mathcal{K}
$$
Properties (i) and (ii) of $Y$ in $\mathcal{K}$ are equivalent and if $\tilde{Y}$ shares either property (i) or (ii) with $Y$, then
$$
|\tilde{Y}-Y|=0 \quad \text { (equivalently, } Y=\tilde{Y}, \text { a.s. })
$$

MA40058 COURSE NOTES ：

Thus
$$
\mathrm{E}\left(S_{n}^{4}\right) \leq n K+3 n(n-1) K \leq 3 K n^{2}
$$
and (see Section 6.5)
$$
\mathrm{E} \sum\left(S_{n} / n\right)^{4} \leq 3 K \sum n^{-2}<\infty
$$
so that $\sum\left(S_{n} / n\right)^{4}<\infty$, a.s., and
$$
S_{n} / n \rightarrow 0, \quad \text { a,s. }
$$