Mixing properties of ARCH and time-varying ARCH processes

There exists very few results on mixing for nonstationary processes. However, mixing is often required in statistical inference for nonstationary processes, such as time-varying ARCH (tvARCH) models. In this paper, bounds for the mixing rates of a stochastic process are derived in terms the conditional densities of the process. These bounds are used to obtain the a, 2-mixing and β-mixing rates of the nonstationary time-varying ARCH(p) process and ARCH(1) process. It is shown that the mixing rate of time-varying ARCH(p) process is geometric, whereas the bounds on the mixing rate of the ARCH(1) process depends on the rate of decay of the ARCH(1) parameters. We mention that the methodology given in this paper is applicable to other processes.