A Central Limit Theorem of Fourier Transforms of Strongly Dependent Stationary Processes

We consider a limiting distribution of the finite Fourier transforms of observations drawn from a strongly dependent stationary process. It is proved that the finite Fourier transforms at different frequencies are asymptotically independent and normally distributed. Our result can apply to a fractional autoregressive integrated moving‐average process and a fractional Gaussian noise, two examples of strongly dependent stationary processes.