Limit theorems for time averages of continuous-state branching processes with immigration

TL;DR

This paper establishes law of large numbers, central limit theorem, and large deviation principles for time averages of subcritical continuous-state branching processes with immigration, under various moment conditions.

Contribution

It provides the first comprehensive limit theorems for time averages of CBI processes, including explicit rate functions for large deviations.

Findings

Proved L^2 law of large numbers for time averages.

Established central limit theorem for the process.

Derived large deviation principle with explicit rate function.

Abstract

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.