  There exist very few results on mixing for non-stationary processes. However,
mixing is often required in statistical inference for non-stationary processes
such as time-varying ARCH (tvARCH) models. In this paper, bounds for the mixing
rates of a stochastic process are derived in terms of the conditional densities
of the process. These bounds are used to obtain the $\alpha$, 2-mixing and
$\beta$-mixing rates of the non-stationary time-varying $\operatorname
{ARCH}(p)$ process and $\operatorname {ARCH}(\infty)$ process. It is shown that
the mixing rate of the time-varying $\operatorname {ARCH}(p)$ process is
geometric, whereas the bound on the mixing rate of the $\operatorname
{ARCH}(\infty)$ process depends on the rate of decay of the $\operatorname
{ARCH}(\infty)$ parameters. We note that the methodology given in this paper is
applicable to other processes.
