Need a justification for the bell-shaped distribution of DCT coefficients.

Question

In my experiments, I have $N=8192$ numbers ($x_n$) which are uniformly distributed on [-128,127]. When I calculate Discrete Cosine Transform (DCT) coefficients ($X_k$) of the numbers and plot the histogram of the coefficients, I see a bell-shaped distribution (probably Gaussian). I need a theoretical justification for this phenomenon. I want to mathematically explain why the coefficients have a bell-shaped distribution. I couldn't prove it by Central Limit Theorem. Could anyone please help me to achieve a detailed justification?

This is the DCT formula: $$ X_k=\sum_{n=0}^{N-1} x_n \cos[\frac{\pi}N (n+\frac{1}2) k] \qquad k=0,...,N-1 $$

This simple Matlab code represents my experiment:

signal_length = 8192;
data = randi([-128, 127], 1, signal_length);
fdata = dct(data);

subplot(211);
hist(data,signal_length);
title('Histogram of data');
subplot(212);
hist(fdata,signal_length);
title('Histogram of DCT coefficients');