4
$\begingroup$

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:

  1. 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.

  2. 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!

  • 0
    By the way it occurs to me that what I call the "usual weak convergence of measures" is probably not what you think of when you think of weak convergence of measures. I mean that in the sense that by the Riesz Representation theorem we can represent bounded (or positive) functionals on the space of continuous functions with compact support as measures. Convergence here is actually different from the probabilistic weak convergence (it is vague convergence), which as I said above is defined for functions which are merely bounded and continuous.2012-02-28

0 Answers 0