I am a graduate student in statistics and am self-studying convergence in probability. I am a little confused on the following problem.
I am trying to follow a proof that claims
$$X_n\sim N(0,1/n) \overset{D}{\rightarrow} 0.$$
The proof uses the definition
$$\lim_{n \to \infty} F_n(t) = F(t)$$
to show the convergence in distribution, where $F(t)$ is c.d.f. of the point mass distribution at $0.$ The proof shows that, for $t<0$,
$$\lim_{n \to \infty} F_n(t) = \mathbb{P}(X_n < t) = 0,$$
and that for $t>0$,
$$\lim_{n \to \infty} F_n(t) = \mathbb{P}(X_n < t) =1.$$
The proof then concludes that $X_n \overset{D}{\rightarrow} 0.$
My question:
I completely understand the steps of the proof and how the limits were found, which is why I chose not to include them here. I do not understand the last statement. Wouldn't it be more correct to say that $X_n \overset{D}{\rightarrow} X$, where $X$ is the point mass distribution at $0$? The definition of convergence of distribution states that a random variable converges to another random variable. $0$ is not a random variable.
Thank you.
Best Answer
A random variable $X$ is a function $X : \Omega \to \mathbb{R}$.
When we write $X=0$ as a random variable, this is just the function taking every point $\omega\in\Omega$ to $X(\omega) = 0 \in \mathbb{R}$.
Note that this is different from the distribution of the random variable $0$, which is the point mass at zero.