Berry–Esseen Bounds and Cramér-Type Moderate Deviations for the Sample Mean and the MLE of the Growth Rate for a Jump-Type CIR Process

Abstract

We study Berry–Esseen bounds and Cramér-type moderate deviations of a jump-type Cox–Ingersoll–Ross (CIR) process driven by a standard Wiener process and a subordinator. In the subcritical case, we obtain the best Berry–Esseen bound of the sample mean and the MLE of the growth rate if the Lévy measure of the subordinator has finite third order moment. Under the Cramér condition, we establish the Cramér-type moderate deviations of the MLE of the growth rate. We first derive a Berry–Esseen bound, a deviation inequality and the Cramér-type moderate deviations for the sample mean of the CIR process by analyzing the asymptotic behaviors of the characteristic function and the moment generating function of the sample mean. Then we analyze a type of additive functional of the jump-type CIR process and use a transformation to study the Berry–Esseen bound and the Cramér-type moderate deviations for the MLE of the growth rate.