Limit theorems for iterated random functions

Abstract

We study geometric-moment contracting properties of nonlinear time series that are expressed in terms of iterated random functions. Under Dini-continuity condition, a central limit theorem for additive functionals of such systems is established. The empirical processes of sample paths are shown to converge to Gaussian processes in the Skorokhod space. An exponential inequality is established. We present a bound for joint cumulants, which ensures the applicability of several asymptotic results in spectral analysis of time series. Our results provide a vehicle for statistical inferences for fractals and many nonlinear time series models.

Asymptotic theory for stationary processes

“…We can view (15) as an iterated random function. The problem of existence of stationary distributions of iterated random functions and the related convergence issues has been extensively studied (Barnsley and Elton (1988), Elton (1990), Duflo (1997), Arnold (1998), Diaconis and Freedman (1999), Steinsaltz (1999), Alsmeyer and Fuh (2001), Jarner and Tweedie (2001), Wu and Shao (2004)). Here we shall present a sufficient condition for (15) so that the representation (1) holds.…”

Nonstationarity-Extended Whittle Estimation

“…For a linear process u t = j∈Z b j ε t−j with ε j being iid, Assumption 2.2 holds if j∈Z |b j | < ∞ and ε 1 ∈ L 4 . For nonlinear processes u t , it is satisfied under a geometric moment contraction (GMC) condition with order 4 [see Wu and Shao's (2004) Proposition 2]. The process {u t } is GMC with order α, α > 0, if there exists a ρ = ρ(α) ∈ (0, 1) such that…”

Oscillations of empirical distribution functions under dependence

“…Then (2.11) has a stationary distribution and X k − X * k α = O(ρ k ) (see [24]). Hence (2.13) holds.…”