Basic problem with convergence in distribution,

Question

Let's define $$P(X_n)=\begin{cases}1,&w.p=\; \frac1n \\0,&w.p = 1-\frac1n\end{cases}$$ I want to ask if that sequence of random variables is converge in distribution. Firstly let's think about cumulative distribution function for sequence $(X_n)$. It's $$F_{X_n}(x)=\begin{cases}0,&x<0 \\1-\frac1n,&x\in[0,1) \\ 1,&x>1\end{cases}$$ And know it's the part im not sure about. If $n \to \infty$ then $1-\frac1n \to 1$. So if $(X_n)$ converges in distribution, it has to converge to a random variable $X$ with cumulative distribution function : $$F_{X}(x)=\begin{cases}0,&x<0 \\ 1,&x\ge0\end{cases}$$ Now for any point that $F_X$ is continuous it has to be : $\lim _{n\to \infty }F_{X_n}(x)=F(x)$. So the only point of discontinuous of $F$ is in $0$, and for any other points the above condition is satisfied. So $X_n$ converges in distribution to $X$. And my question is : Am i thinking correctly ?

Answer 1

In general $Y_n\stackrel{p}{\to}c\iff Y_n\stackrel{d}{\to}c$. Note that in your case the random variables converge in probability to $0$. It follows that they converge in distribution to zero too.