  The interplay of the tunneling transfer of charges and the emission and
absorption of light can be investigated in a setup, where a voltage-biased
Josephson junction is connected in series with a microwave cavity. We focus
here on the emission processes of photons and analyze the underlying
time-dependent statistics using the second-order correlation function
$g^{(2)}(\tau)$ and the waiting-time distribution $w(\tau)$. Both observables
highlight the crossover from a coherent light source to a single-photon source.
Due to the nonlinearity of the Josephson junction, tunneling Cooper pairs can
create a great variety of non-classical states of light even at weak driving.
Analytical results for the weak driving as well as the classical regime are
complemented by a numerical treatment for the full nonlinear case. We also
address the question of possible relations between $g^{(2)}(\tau)$ and
$w(\tau)$ as well as the specific information which is provided by each of
them.
