Limit theorems for locally stationary processes
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Limit theorems for locally stationary processes Rafael Kawka1 Received: 10 September 2019 / Revised: 10 June 2020 © The Author(s) 2020
Abstract We present limit theorems for locally stationary processes that have a one sided timevarying moving average representation. In particular, we prove a central limit theorem (CLT), a weak and a strong law of large numbers (WLLN, SLLN) and a law of the iterated logarithm (LIL) under mild assumptions using a time-varying Beveridge– Nelson decomposition. Keywords Locally stationary process · Central limit theorem · Law of large numbers · Law of the iterated logarithm
1 Introduction In this paper we consider locally stationary processes, defined via a triangular sequence of stochastic processes {ηt,T }t=1,...,T with T ∈ N, where every ηt,T has a representation of the form ηt,T = μ
∞ t + ψ j,t,T εt− j , t = 1, . . . , T . T
(1)
j=0
Throughout this paper we impose the following assumption on the error sequence {εt }t∈Z , on the moving average coefficients ψ j,t,T and on the trend function μ. Assumption 1.1 The random variables {εt }t∈Z are independent and identically distributed with Eεt = 0, Eεt2 = 1 and E|εt |2+κ < ∞ for some κ > 0. The coefficients ψ j,t,T in the moving average representation (1) fulfill sup |ψ j,t,T | ≤ t,T
B 1
K , l( j)
Rafael Kawka [email protected] Fakultät Statistik, Technische Universität Dortmund, Vogelpothsweg 87, 44227 Dortmund, Germany
123
R. Kawka
with constant K independent of T and some positive deterministic sequence {l( j)} j∈N0 satisfying ∞ j < ∞. l( j) j=0
The trend function μ : [0, 1] → R is assumed to be bounded and continuous almost everywhere. Remark 1.2 In contrast to the definition of Dahlhaus and Polonik (2006) we restrict locally stationary processes to have a one-sided moving average representation. Nonetheless, our definition covers most of the important examples of locally stationary processes. For instance, it follows from Dahlhaus and Polonik (2009, Proposition 2.4) that time-varying causal ARMA processes have a representation of the form (1). The idea behind locally stationary processes is that, after rescaling the time domain to the unit interval, the process can be approximated locally in time by a stationary process. Therefore, one usually assumes that ψ j,t,T ≈ ψ j (t/T ) for some well behaving functions ψ j . Assumption 1.3 There exist functions ψ j : [0, 1] → R with K , l( j) K V (ψ j ) ≤ l( j)
ψ j ∞ ≤
and T ψ j,t,T − ψ j t ≤ K , for all T ∈ N, T l( j)
(2)
t=1
where V ( f ) denotes the total variation of a function f on [0, 1]. Remark 1.4 The coefficient functions are uniquely defined almost everywhere. To see this let {ηt,T }t=1,...,T be locally stationary process with moving average coefficients ψ j,t,T and corresponding coefficient functions ψ j . Let φ j be another set of coefficient functions that fulfills Assumption 1.3. Then it holds that T t t 1 − φ ψ j j T →∞ T T T t=1 T T 1 t ψ j,t,T − φ j t + ≤ lim
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