How does the power spectrum of the amplitude envelope depend on its underlying carrier frequency?

when looking at the amplitude envelope of an oscillation (see red trace in Figure below), it seems to me as if the frequency content of the envelope depends on the the frequency of the underlying carrier signal (blue trace). More specifically, it looks as if the fastest fluctuation the envelope can exhibit is related to the frequency of the carrier signal. There should be a straightforward mathematical relation between the carrier frequency and the fastest frequency in the envelope. If I think about the individual peaks of the oscillation as "samples" for the envelope, then the fastest frequency the envelope contains should be carrier_freq/2, but I'm not sure if this is correct. I'd appreciate any help or hints in the right direction.