Some general properties of photon correlations are discussed in a simple way through an analysis of the two-detector measurement scheme. It is shown that the assumption of the discreteness of the random process leads directly to the conclusion that the zero-delay value of the correlation function is only bound to be non-negative. The adopted approach allows discussing in a more intuitive way the photon correlation properties of different optical fields, including non-classical fields presenting an apparent violation of the Cauchy-Schwarz inequality. The comparison between the two- and the single-detector experiment clarifies the role of the operator ordering in the definition of the correlation function.

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