A Cauchy-Schwarz inequality in photon noise spectra is derived and discussed for macroscopic fields of light. Violation of this Cauchy-Schwarz inequality is demonstrated experimentally in quantum noise-correlated light beams radiated from a pair of electrically coupled light-emitting diodes. Our measurements directly demonstrate that the violation of this Cauchy-Schwarz inequality can be observed in the absence of sub-shot noise fluctuations in either the single-beam intensity or the dual-beam intensity difference. Therefore, the dual-beam quantum correlation condition required for the violation of this Cauchy-Schwarz inequality is not associated with sub-shot noise fluctuations.