Casella & Berger define convergence in distribution as follows:
$X_n \rightarrow X $ (in distribution) if
$$\lim_{n\rightarrow \infty} F_{X_n}(x)=F_X(x)$$ for all $x$ such that $F_X(x)$ is continuous.
This resembles pretty much the concept of pointwise convergence of a sequence of functions, so is it fair to say that if $\{F_{X_n}\}$ converges pointwise to $F_X$ then $\{X_n\}$ converges in distribution to $X$? (the converse may not be true since the convergence in distribution requires only convergence in the continuity points).
Is my reasoning ok?
Thanks in advance.
Best Answer
Pointwise convergence gives convergence at every point. In particular we have convergence at continuity points so we surely have $X_n\to X$ in distribution. To show that the converse is not true take $X_n =\frac 1 n, X=0$ and consider convergence at $0$.