pdf of the sum of gaussian distributions using fft Ask
The FFT algorithm expects the origin of the signal to be on the leftmost sample of the signal. You are convolving two shifted Gaussian together, yielding a Gaussian that is even more shifted. Because of the FFT imposes periodicity, when the curve is shifted past the right edge it comes back in on the left edge.

Whit an odd number the shift is exactly the full width of the signal, so you don’t notice it. 

A solution would be to use a zero-mean Gaussian distribution, which you can obtain from yours by ifftshift. After performing the convolution, you can shift it back to where it was using fftshift.

np.fft.fftshift(ifft(fft(np.fft.ifftshift(pdf))**2))