A new theorem on the existence of invariant distributions with applications to ARCH processes

Abstract

A new theorem on the existence of an invariant initial distribution for a Markov chain evolving on a Polish space is proved. As an application of the theorem, sufficient conditions for the existence of integrated ARCH processes are established. In the case where these conditions are violated, the top Lyapunov exponent is shown to be zero.

Ergodic decompositions associated with regular Markov operators on Polish spaces

“…Szarek and coworkers [20,24,27,28] and Ollivier [25]. More specific examples of Markov operators on Polish spaces are given by iterated function systems [26,23], ARCH processes in econometrics [19] and random dynamical systems on separable Banach spaces [16,15], which also have various applications in mathematical biology.…”

On Asymptotic Theory for ARCH (∞) Models

“…Douc et al () provide sufficient conditions for the existence of a stationary causal solution of an ARCH( ) process, which allows coefficients to decay at a power law and hence includes the FIGARCH and fractional GARCH models. In the FIGARCH case, this solution necessarily implies an infinite variance of the process, as shown by Kazakevičius and Leipus () and Kazakevičius and Leipus ().…”

Stationary Integrated Arch(∞) and Ar(∞) Processes With Finite Variance

“…1 However, the existence of a stationary solution of the LM(d)-ARCH equation in (1.4) with finite fourth moment was not rigorously established and the validity of (1.5) remained open. See Davidson (2004), Giraitis et al (2000a), Kazakevičius and Leipus (2003), Stȃricȃ (2000, 2003) for a discussion of controversies surrounding the FIGARCH and the LM(d)-ARCH models.…”

“…Condition (1.8) rules out integrated GARCH( p, q) as well as any IARCH(∞) models with sufficiently fast decaying lags which are known to admit a stationary solution with infinite variance, see Kazakevičius and Leipus (2003), Douc, Roueff, and Soulier (2008), Robinson and Zaffaroni (2006). It turns out that covariance stationary solutions of (1.7) always have long memory, in the sense that the covariance function is nonsummable and the spectral density is infinite at the origin, see Corollary 2.1.…”