  In this work we investigate limit theorems for the time-averaged process
$\left(\frac{1}{t}\int_0^t X_s^x ds\right)_{t\geq 0}$ where $X^x$ is a
subcritical continuous-state branching processes with immigration (CBI
processes) starting in $x \geq 0$. Under a second moment condition on the
branching and immigration measures we first prove the law of large numbers in
$L^2$ and afterward establish the central limit theorem. Assuming additionally
that the big jumps of the branching and immigration measures have finite
exponential moments of some order, we prove in our main result the large
deviation principle and provide a semi-explicit expression for the good rate
function in terms of the branching and immigration mechanisms. Our methods are
deeply based on a detailed study of the corresponding generalized Riccati
equation and related exponential moments of the time-averaged process.
