I know the convergence in distribution and the weak convergence. but I have two questions:
First one: does weak convergence implies pointwise convergence or it is the same?
And second one: I have the following definition of weak convergence: A sequence of distribution functions $(F_{n})_{_{n\geq 1}}$ converges weakly against a distribution function $F$ if $\lim_{n \to \infty}F_{n}(x)=F(x)$ for all continuity points of $F$. My question: What is if I want to show the weak convergence for a family of distribution functions:$(F_{t})_{_{t >0}}$ ($t$ is a real number) i have two ideas: first one: i thought that i can choose $[t]=$greatest integer less than or equal to $t$, instead of $t$ . But if this is possible, i don't know why...
An my second idea, i show the weak convergence for all sequences $t_{n}$, which converge to $\infty$.
Thanks for your help
Define $F_{c}:\mathbb{R}\rightarrow\left[0,1\right]$ by $x\mapsto1$ if $x\geq c$ and $x\mapsto0$ otherwise.
If $c_{n}$ converges to $c$ then $F_{c_{n}}$ converges to $F_{c}$ weakly. That is $F_{c_{n}}\left(x\right)\rightarrow F_{c}\left(x\right)$ for each $x\neq c$ (i.e. all continuity points of $F_{c}$)
Pointwise convergence is there if $F_{c_{n}}\left(x\right)\rightarrow F_{c}\left(x\right)$ for each $x$, so also for $x=c$.
However $F_{c_{n}}\left(c\right)\rightarrow F_{c}\left(c\right)=1$ askes for $c\geq c_{n}$ for $n$ large enough wich is not a consequence of $c_{n}\rightarrow c$.
E.g if $F_{n^{-1}}$ converges weakly to $F_{0}$ while $F_{n^{-1}}\left(0\right)=0$ for each $n$ and $F_{0}\left(0\right)=1$. So there is no convergence $F_{n^{-1}}\left(0\right)\rightarrow F_{0}\left(0\right)$.
addendum
Let $F$ be a CDF and denote $C:=\left\{ x\in\mathbb{R}\mid F\text{ is continuous at }x\right\} $.
A family $\left(F_{t}\right)_{t>0}$ of CDF's converges weakly to $F$ under $t\rightarrow a$ if: $$\lim_{t\rightarrow a}F_{t}\left(x\right)=F\left(x\right)\text{ for each }x\in C$$
This is the case if and only $F_{t_{n}}$ converges weakly to $F$ for every sequence $\left(t_{n}\right)_{n}$ that converges to $a$.