The functional central limit theorem and structural change test for the HAR(∞) model

•Some powered process of the hyperbolic-memory HAR model is considered.•Sufficient conditions for L2-NED of the process are given.•The FCLT for the process is proved.•The Limiting distribution of the CUSUM statistics of the process is derived.

In this paper, we study the functional central limit theorem (FCLT) for the infinite-order heterogeneous autoregressive model of realized volatility (HAR(∞)). We prove under proper assumptions that the process is L2-NED and then obeys the FCLT. As an application of the FCLT, we derive the limit distribution of the CUSUM statistics to detect the structural change of the model.