What does $L^2$ on an arrow mean?

Question

This post says

If the $n$ observations in the sample $$\xi_n = [x_1, \dots, x_n]$$ are the realizations of n mutually independent random variables $X_1 \dots X_n$ having the distribution function $F_X(x)$, then

$$F_n(x)\stackrel{L^2}\to F(x)$$

for any $x \in R$

What does $L^2$ on an arrow mean? Does that mean "goes to" under some condition? If yes, what condition does the $L^2$ mean?

Answer 1

Yes, it means it goes to something, i.e. it converges, and the some condition here is the $L^2$ -norm. $L^2$ is a special case of the so-called $L^p$-spaces. Functions in $L^2$ are sometimes called square integrable functions as for $f\in L^2(\Bbb R)$ we have that

$$\int_{-\infty}^\infty |f(x)|^2\mathrm dx < \infty$$

The $L^2$-norm is defined in a similiar manner, i.e.

$$\|f\|_{L^2(\Bbb R)}:=\int_{-\infty}^\infty |f(x)|^2\mathrm dx$$

Answer 2

It means convergence in the $L^2$-norm, that is, $$ \| F_n(x)-F(x)\|_{ L^2(\mathbb{R}) }\to 0 $$ as $n\to \infty $.