Invariance principles for fractionally integrated nonlinear processes

Abstract

We obtain invariance principles for a wide class of fractionally integrated nonlinear processes. The limiting distributions are shown to be fractional Brownian motions. Under very mild conditions, we extend earlier ones on long memory linear processes to a more general setting. The invariance principles are applied to the popular R/S and KPSS tests.Comment: Published at http://dx.doi.org/10.1214/074921706000000572 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

An Invariance Principle for Fractional Brownian Sheets

“…[26] and references therein) to random fields ( [14]). Note that Wu and Shao [26] also established an invariance principle for the socalled Type II fractional integrated processes, which are slightly different from our model (1). Theorem 3 follows as usual from the convergence of finite-dimensional distributions and tightness (see e.g.…”

Projective Stochastic Equations and Nonlinear Long Memory

“…This completes the proof. Here ξ q := E 1/q |ξ | q and q = 2 for 0 < d < 1 2 ; see [25,Theorem 2.1], and also [23] and [24]. The above-mentioned papers verify (65) for several classes of Bernoulli shifts.…”

“…and, hence, (2) = ∞ for 0 < d < 1 2 . The above argument suggests that projective moving averages posses a different 'memory mechanism' from the fractionally integrated processes in [25].…”

“…(i) Let us show that the g k−t,t as defined in (25) satisfy ∞ k=0 Eg 2 t−k,t < ∞. From (22) and (25), we have the recurrent inequality…”

“…Baillie and Kapetanios [1] considered the fractionally integrated process with nonlinear autoregressive innovations. A general invariance principle for fractionally integrated models with weakly dependent innovations satisfying the projective dependence condition of Wu [23] is established in [25]. See also [24] and Remark 5 below.…”