Abstract: In the present paper we consider the main probabilistic properties of the Markov chain Xt=aXt-1+[a0+(a1 +(Xt-1)++a1-(Xt-1) -)2β]1/2εt , that we call the β-ARCH model. We examine the inevitability, irreducibility, Harris recurrence, ergodicity, geometric ergodicity, α-mixing, existence and nonexistence of finite moments and exponential moments of some order and sharp upper bounds for the tails of the stationary density of the process {Xt} in terms of the common density of the εt's.
Key words and phrases: Markov chain, invertibility, ergodicity, mixing, tail of the stationary density, ARCH model, nonlinear time series, autoregressive.