## Bernoulli

• Bernoulli
• Volume 17, Number 1 (2011), 320-346.

### Mixing properties of ARCH and time-varying ARCH processes

#### Abstract

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 $α$, 2-mixing and $β$-mixing rates of the non-stationary time-varying ARCH($p$) process and ARCH($∞$) process. It is shown that the mixing rate of the time-varying ARCH($p$) process is geometric, whereas the bound on the mixing rate of the ARCH($∞$) process depends on the rate of decay of the ARCH($∞$) parameters. We note that the methodology given in this paper is applicable to other processes.

#### Article information

Source
Bernoulli, Volume 17, Number 1 (2011), 320-346.

Dates
First available in Project Euclid: 8 February 2011

https://projecteuclid.org/euclid.bj/1297173845

Digital Object Identifier
doi:10.3150/10-BEJ270

Mathematical Reviews number (MathSciNet)
MR2797994

Zentralblatt MATH identifier
1284.62550

#### Citation

Fryzlewicz, Piotr; Subba Rao, Suhasini. Mixing properties of ARCH and time-varying ARCH processes. Bernoulli 17 (2011), no. 1, 320--346. doi:10.3150/10-BEJ270. https://projecteuclid.org/euclid.bj/1297173845

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