It is known that if $X_i$, $i=1,\dots,n$, are i.i.d random variables with mean $0$ and variance $1$, then $\frac {1} {\sqrt n} (X_1+ \dots+X_n)$ converges in distribution to a $\mathcal N(0,1)$ (CLT).
What if we have $\{ X_t : t \in [0,n]\}$ i.i.d random variables with mean $0$ and variance $1$ ? Is there a similar result to the classical CLT for the asymptotic distribution of
$$\frac {1} {\sqrt n} \int_0^nX_t \, dt$$
This is a soft question as I am not sure if what I am asking makes sense.