There is something I don't understand about the central limit theorem in the frequancy domain.
The central limit theorem tells us that if we have $n$ probability density functions $f_i$ then $$h=f_1*\cdots*f_n$$ approches a gaussian distrabution, but let's say $f=f_1=f_2=\cdots=f_n$ that means the convolution of all of them in the frequancy domain is $$F\{h\}=F\{f\}^n$$ so if the ratio between the frequancy with the largest amplitude to the second largest amplitude in $f$ is $r$ then in $h$ it is $r^n$ which is infinite. So that means $h$ has only one frequancy so it's a sinusodial not a gaussian. What am I missing?
Recall $h$ is the density of $X_1+\dots X_n$ (where $X_i \overset{\text{i.i.d}}{\sim} f$, but the CLT deals with $(X_1+\dots X_n)/n$, provided of course that $X_i$ are centered and have variance 1. The FT of the density of this is $$\left( F \{ f \} \left ( \frac{k}{n} \right)\right)^n$$ And so if the ratio between 'the largest amplitude' and 'the second largest amplitude' in $F\{f\} $ is $\lambda$, say, then this ratio becomes $(\lambda/n)^n$ which tends to 0 as $n$ tends to infinity.