  We discuss some mathematical aspects of photon antibunching and sub-Poisson
photon statistics. It is known that Bell's inequalities for entangled states
can be reduced to the Cauchy-Bunyakovsky inequalities. In this note some
rigorous results on impossibility of classical hidden variables representations
of certain quantum correlation functions are proved which are also based on the
Cauchy-Bunyakovsky inequalities.
  The difference K between the variance and the mean as a measure of
non-classicality of a state is discussed. For the classical case K is
nonnegative while for the n-particle state it is negative and moreover it
equals -n. The non-classicality of quantum states discussed here for the
sub-Poisson statistics is different from another non-classicality called
entanglement though both can be traced to the violation of the
Cauchy-Bunyakovsky inequality.
