On some 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 strong and 2-mixing rates of the nonstationary time-varying ARCH$(p)$ process and ARCH$(\infty)$ 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$(\infty)$ process depends on the rate of decay of the ARCH$(\infty)$ parameters. These results are generalised to consider mixing rates for moving averages of ARCH$(\infty)$ random variables.