# 概率学|Probability代写 MATH 605

Let $F_{n}$ be a sequence of DFs defined by
$$F_{n}(x)= \begin{cases}0, & x<0 \ 1-\frac{1}{n}, & 0 \leq x<n \ 1, & n \leq x .\end{cases}$$
Clearly $F_{n} \stackrel{w}{\rightarrow} F$, where $F$ is the DF given by
$$F(x)= \begin{cases}0, & x<0 \ 1, & x \geq 0\end{cases}$$

Note that $F_{n}$ is the DF of the RV $X_{n}$ with PMFand $F$ is the DF of the RV $X$ degenerate at 0 . We have
$$E X_{n}^{k}=n^{k}\left(\frac{1}{n}\right)=n^{k-1}$$
where $k$ is a positive integer. Also $E X^{k}=0$. So that
$$E X_{n}^{k} \nrightarrow E X^{k} \quad \text { for any } k \geq 1$$

## MATH605 COURSE NOTES ：

Proof. Since $X$ is an RV, we can, given $\varepsilon>0$, find a constant $k=k(\varepsilon)$ such that
$$P{|X|>k}<\frac{\varepsilon}{2} .$$
Also, $g$ is continuous on $\mathcal{R}$, so that $g$ is uniformly continuous on $[-k, k]$. It follows that there exists a $\delta=\delta(\varepsilon, k)$ such that
$$\left|g\left(x_{n}\right)-g(x)\right|<\varepsilon$$
whenever $|x| \leq k$ and $\left|x_{n}-x\right|<\delta$. Let