My question is convergence in distribution seems to be defined differently in Wikipedia and in Kai Lai Chung's book. My view is that the one by Wikipedia is a standard definition of convergence in distribution, and the one by Chung is actually vague convergence not convergence in distribution. I wonder if the two definitions are equivalent, and why?
Following are relevant quotes from the two sources:
From Wikipedia
A sequence $\{X_1, X_2, …\}$ of random variables is said to converge in distribution to a random variable $X$ if $ \lim_{n\to \infty} F_n(x) = F(x)$ for every number $x ∈ \mathbb{R}$ at which $F$ is continuous. Here $F_n$ and $F$ are the cumulative distribution functions of random variables $X_n$ and $X$ correspondingly.
From Kai Lai Chung's A course in probability theory, consider (sub)probability measures (s.p.m.'s or p.m.'s) on $\mathbb{R}$.
definition of convergence "in distribution" (in dist.)- A sequence of r.v.'s $\{X_n\}$ is said to converge in distribution to d.f. $F$ iff the sequence $\{F_n\}$ of corresponding d.f.'s converges vaguely to the d.f. $F$.
My rephrase of vague convergence of a sequence of distribution functions (d.f.'s) based on the same book is
We say that $F_n$ converges vaguely to $F$, if their s.p.m.'s are $\mu_n$ and $\mu$, and $\mu_n$ converges to $\mu$ vaguely.
On p85 of Chung's book, vague convergence of a sequence of s.p.m.'s is defined as
a sequence of subprobability measures (s.p.m.'s) $\{ \mu_n, n\geq 1 \}$ is said vaguely converge to another subprobablity measure $\mu$ on $\mathbb{R}$, if there exists a dense subset $D$ of the real line $\mathbb{R}$ so that $ \forall a \text{ and } b \in D \text{ with } a .
Thanks and regards!