Asymptotic behavior of frequency polygon estimation of density function for dependent samples

This paper pays attention to the frequency polygon, which is constructed by connecting with straight lines the mid-bin values of a histogram. As a density estimator based on the histogram technique, the frequency polygon has the advantage of computational simplicity and has been widely used in many fields. The purpose of this article is to investigate the weak consistency, the uniformly weak consistency, and the rate of the uniformly weak consistency for frequency polygon estimation of the density function under $\alpha$ -mixing samples, which improve and extend the corresponding ones in the literature. In addition, the simulation study and real data analysis are also presented to verify the validity of the theoretical results based on finite samples.