What exactly is Cramer-Wold device? (What's the difference b/w Cramer-Wold theorem and Cramer-Wold device?)
By Emma Terry •
I'm a bit confused about the exact statement of the Cramer-Wold device. I've read multiple sources which seem to be giving different definitions (the first one is more common).
- Random vectors $X_n \in \mathbb{R}^d$ satisfy $X_n \rightarrow_d X$ if and only if $\langle a, X_n \rangle \rightarrow_d \langle a, X \rangle$ for all $a \in \mathbb{R}^d$.
(Source: Proposition 2.7)
- If $\langle a, X_n \rangle \rightarrow_d \langle a, X \rangle$ for all $a \in \mathbb{R}^d$ with $\|a\| = 1$, then $X_n \rightarrow_d X$.
(Source: Theorem 13)
Also, What's the difference between Cramer-Wold theorem and Cramer-Wold device? Is there a difference?
$\endgroup$ 4 Reset to default
