What is the $n$th convolution of a function $f:\mathbb R\rightarrow \mathbb R$ with itself, $f*f*\cdots*f$ in the $n\rightarrow \infty$ limit? Since the Gaussian is a fixed point of convolutions with appropriate rescaling/normalization, is there a general result analogous to the central limit theorem here?
For example the $n$th convolution of a rectangle pulse $\chi$ with itself is the Fourier transform of $\sin^n x / x^n$, which seems to approach a Gaussian with an appropriate normalization. First, is this true, and if so, is this true more generally for a larger class of functions?