Entropies of Cox–Ingersoll–Ross and Bessel processes as functions of time and of related parameters

We investigate the long-time asymptotic behavior of various entropy measures associated with the Cox–Ingersoll–Ross (CIR) and squared Bessel processes. As the one-dimensional distributions of both processes follow noncentral chi-squared laws, we first derive sufficient conditions for the existence of these entropy measures for a noncentral chi-squared random variable. We then analyze their limiting behavior as the noncentrality parameter approaches zero and apply these results to the Cox–Ingersoll–Ross and squared Bessel processes. We prove that, as time tends to infinity, the entropies of the CIR process converge to those of its stationary distribution, while for the squared Bessel process, the Shannon, Rényi, and generalized Rényi entropies diverge, however, the Tsallis and Sharma–Mittal entropies may diverge or remain finite depending on the entropy parameters. Finally, we demonstrate that, as the CIR process converges to the squared Bessel process, the corresponding entropies also converge.