Limit theorems for locally stationary processes

We present limit theorems for locally stationary processes that have a one sided time-varying 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.


Introduction
In this paper we consider locally stationary processes, defined via a triangular sequence of stochastic processes {η t,T } t=1,...,T with T ∈ N, where every η t,T has a representation of the form Throughout this paper we impose the following assumption on the error sequence {ε t } t∈Z , on the moving average coefficients ψ j,t,T and on the trend function μ.

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 .
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 {η t,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 For every u ∈ [0, 1] we define the process {η t (u)} t∈Z via By Assumption 1.3 the centered process {η t (u) − μ(u)} t∈Z is weakly stationary with long-run variance given by Ψ 2 (u), where The main purpose of the process {η t (u)} t∈Z is to approximate {η t,T } t=1,...,T . In particular, the process {η t,T } t=1,...,T should approximately behave like the stationary process {η t (u)} t∈Z in the rescaled time point u = t/T . For brevity, we define the auxiliary Under the stated assumptions it holds that (cf. Lemma A.1 in the appendix) as T → ∞. Hence, the process {η t (u)} t∈Z approximates the locally stationary process on average over all rescaled time points 1/T , 2/T . . . , 1. Later we will strengthen condition (2) in order to obtain a stronger approximation.

Remark 1.5
The construction of locally stationary processes with time dependent moving-average coefficients ψ j,t,T on the one hand and approximating functions ψ j on the other hand looks cumbersome at first glance. It seems more natural to define locally stationary processes directly via (3). However, it was already pointed out by Künsch (1995) and Dahlhaus and Polonik (2009) that this rules out interesting examples such as time-varying autoregressive processes.
Consider the stationary approximating process {η t (u)} t∈Z for some fixed u ∈ [0, 1]. By the Beveridge-Nelson decomposition (cf. Phillips and Solo 1992) it holds that which is well defined due to Assumption 1.3. Setting u = t/T we obtain a time-varying Beveridge-Nelson decomposition for the auxiliary process {η t,T } t=1,...,T . Lemma 1.6 (Time-varying Beveridge-Nelson decomposition) The auxiliary process {η t,T } t=1,...,T exhibits a representation of the form The time-varying Beveridge-Nelson decomposition will be useful for the derivation of the main results in this paper. In particular, we will use it to generalize the proof techniques of Phillips and Solo (1992) to the locally stationary framework.

Main results
The first limit theorem we present is a CLT for locally stationary processes. To motivate the outcome, we first derive the result for an easy example. Let {η t,T } t=1,...,T be defined by for some bounded variation function φ : [0, 1] → R and {ε t } t∈Z being a sequence of independent and identically N (0, 1) distributed random variables. Then, it holds that Since φ is of bounded variation it is square-integrable on the unit interval and it holds that and Lévy's continuity theorem implies that Note that the approximating stationary process {η t (u)} t∈Z is defined by η t (u) = φ(u)ε t with long-run variance given by φ 2 (u). Hence, the variance of the limiting distribution in (4) is equal to the integrated long-run variance of the auxiliary process. This result also holds for arbitrary centered locally stationary processes.
..,T be a locally stationary process with movingaverage representation (1) that satisfies Assumptions 1.1 and 1.3. Then, as T → ∞, it holds that where Ψ L 2 denotes the L 2 norm of Ψ on the unit interval.
Proof It suffices to show the claim for the auxiliary process {η t,T } t=1,...,T since and the second term goes to zero in probability by Lemma A.1. Without loss of generality we assume that μ(u) = 0 for all u ∈ [0, 1]. By Lemma 1.6 it holds that We show that the first term in (5) converges in distribution and the second term vanishes in probability. By the i.i.d. assumption on the innovation terms it holds that Next, we verify the Lyapunov condition. By Assumption 1.1 there exists some κ > 0 such that E|ε t | 2+κ is finite. Hence, From the Lindeberg-CLT for triangular arrays we deduce that To finish the proof it remains to show that the second term in (5) goes to zero in probability. It holds that Taking partial sum of the first term and dividing by T 1/2 leads to: as the sum over t is telescopic. Since for an arbitrary t ∈ {1, . . . , T } it follows that the term on the right hand side of (7) converges to zero in probability.
It remains to prove that the scaled partial sum of the second term in (6) also vanishes asymptotically. It holds that which converges to zero if the V (ψ j ) are summable. Using the definition of the total variation we obtain which is finite by Assumption 1.
In order to prove a SLLN and a LIL we require a stronger assumption that connects the coefficient functions ψ j,t,T and the approximating functions ψ j . The following assumption, that immediately implies condition (2), corresponds to assumption (69) in Dahlhaus (2012).

Assumption 2.3
The functions ψ j and the moving average coefficients ψ j,t,T satisfy , for all T ∈ N.
Previously, we observed that the stationary process {η t (u)} t∈Z approximates the locally stationary process on average over the series. Under Assumption 2.3 we have a better approximation as it now holds that η t,T −η t,T = O P (T −1 ). This follows from the fact that Consequently, the stationary process {η t (u)} t∈Z approximates the locally stationary process {η t,T } t=1,...,T in every rescaled time point u = t/T . In fact, we even have the following strong approximation. It holds that as T → ∞ (cf. Lemma A.2 in the appendix). The trend function μ is assumed to be bounded and continuous everywhere. This implies that it is Riemann integrable and we immediately deduce that It remains to show that Using Lemma 1.6 we first need to verify that It holds that sup u∈(0,1) Since the ε t 's are independent and identically distributed with E(ε 1 ) = 0 and Eε 2 1 < ∞ almost sure convergence of (9) follows from Cuzick (1995, Theorem 1.1) or Choi and Sung (1987, Theorem 5).
It remains to show that It holds that Hence, the term in (10) converges sufficiently fast to zero in probability and almost sure convergence follows from the Borel-Cantelli lemma. The proof of (11) is identical. At last we have to show (12). It holds that In contrast to the proof of Theorem 2.4 it is not sufficient to investigate the second moments of these terms, as d −1 T decays too slowly. However, we adapt the proof using fourth moments. For the term in (13) and, by the Borel-Cantelli lemma, almost sure convergence. The claim in (14) is proven in exactly the same way. To show (15) consider The claim follows by the same arguments as above.
Obviously, if the coefficients ψ j,t,T are not time-dependent, the statement of Theorem 2.5 coincides with the LIL for linear processes which was proven by Phillips and Solo (1992). Proof It holds that Using Hölder's inequality we obtain

This implies that
since, by Assumption 1.1, {ε t } t∈Z is a sequence of independent random variables and there exists some κ > 0 such that E|ε t | 2+κ < ∞. Hence, for all α > 0 we obtain, applying the Bonferroni and the Markov inequalities, which implies the claim.