  In a process called parametric down-conversion, a dc-biased Josephson
junction coupled to two microwave resonators emits photon pairs when the
Josephson frequency matches the sum of the two resonance frequencies. Recent
experiments have shown that such a setup permits analyzing the correlation of
the radiation. Motivated by these results, we study theoretically the full
counting statistics of a non-degenerate parametric oscillator below the
threshold of instability. Furthermore, we analyze the second-order coherences
of the radiation and discuss thermal effects on the radiation statistics. We
provide results for the driving strength at which the Cauchy-Schwarz inequality
is most strongly violated. Additionally, we study the impact of asymmetry in
the linewidth of the modes---a distinctive property of the non-degenerate
resonance effect. In particular, we find that the radiation from the mode with
the larger linewidth preferably takes the total detuning, while the other mode
emits photons at its resonance frequency.
