Asymptotic normality of a kernel-type estimator for periodic Poisson intensity with power trend

The nonhomogeneous Poisson process is widely applied to model random events with time-varying intensity. Previous studies have introduced kernel-based nonparametric estimators for cyclic Poisson processes and extended them to include linear, quadratic, and power function trends. While asymptotic properties such as bias, variance, and consistency have been established, the asymptotic distribution of these estimators remains underexplored. This paper addresses this gap by proving the asymptotic normality of the kernel-type estimator for intensity functions modeled as the product of a periodic component and a power trend. Using the Central Limit Theorem for independent but non-identically distributed variables, we derived the limiting distribution under various conditions on the trend parameter. These results provide a theoretical foundation for constructing confidence intervals and hypothesis tests in nonparametric inference for nonhomogeneous Poisson processes.