这是一份warwick华威大学ST231-10的成功案例

Suppose $Y(k)=m(k)+X(k)$ with a deterministic function $m(k)$ and $X(k)$ is a stochastic process with all moments $\mathbb{E}\left[|X(k)|^{p}\right]$ existing for $k \in \mathbb{N}$ and distributions $F_{k}(x):=$ $\operatorname{Prob}({\omega \in \Omega: X(k, \omega) \leq x})$ symmetric to the origin. Then the following are equivalent:

1. For each $p \in \mathbb{N}$ it holds:
   $$
   \mathbb{E}\left[|Y(k)-\mathbb{E}[Y(k)]|^{p}\right]=c(p) \cdot \sigma^{p}|k|^{p H}
   $$
2. For each $k$ the following functional scaling law holds on $\operatorname{Sym}{0}^{0}(\mathbb{R})$ : $$ F{k}(x)=F_{1}\left(k^{-H} x\right)
   $$
   where $S y m C_{0}^{0}(\mathbb{R})$ is the set of symmetric (with respect to the $y$-axis) continuous functions with compact support.

ST231-10  COURSE NOTES ：

$$
\begin{aligned}
\sigma_{\log (x+)} &=\frac{1}{x_{l}} \cdot\left(\frac{q \cdot(1-q)}{n \cdot\left(f\left(x_{l}\right)\right)^{2}}\right)^{\frac{1}{2}} \
&=\sqrt{\frac{q \cdot(1-q)}{n}} \cdot \frac{1}{x_{l} \cdot f\left(x_{l}\right)} .
\end{aligned}
$$
For example, if $X \sim \mathcal{N}\left(0, \sigma^{2}\right)$, the propagation of the error can be written as
$$
\begin{aligned}
\sigma_{\log (x)} &=\sqrt{\frac{q \cdot(1-q)}{n}} \cdot \frac{\sqrt{2 \pi} \cdot \sigma}{x \cdot \exp \left(-\frac{x^{2}}{2 \sigma^{2}}\right)} \
&=\sqrt{\frac{q \cdot(1-q)}{n}} \cdot \frac{\sqrt{2 \pi}}{y \cdot \exp \left(-\frac{y^{2}}{2}\right)}
\end{aligned}
$$