Lecture 14. G) Somewhat Pedantic Remark on Notation
Somewhat Pedantic Remark on Notation
You may have noticed that statements like
[math]\sqrt{n}\left(X_{n}-\mu\right)\overset{d}{\rightarrow}N\left(0,\sigma^{2}\right)[/math]
seem overly complicated. It may seem simpler to write the statement above as
[math]\frac{\sqrt{n}\left(X_{n}-\mu\right)}{\sigma}\overset{d}{\rightarrow}N\left(0,1\right)[/math]
or even
[math]X_{n}\overset{d}{\rightarrow}N\left(\mu,\frac{\sigma^{2}}{n}\right)[/math].
While the second statement is perfectly fine, the last statement is problematic. The reason is that the right-hand side, [math]N\left(\mu,\frac{\sigma^{2}}{n}\right)[/math], is the distribution limit of the left-hand side in [math]n[/math], and so should not depend on [math]n[/math]. So, try to keep [math]n[/math] in the left-hand side. If you really want to have it on the right-hand side, you can use the alternative notation,
[math]X_{n}\overset{\sim}{\sim}N\left(\mu,\frac{\sigma^{2}}{n}\right).[/math]
