2.1. Locally stationary time series in discrete time

We follow the concept of local stationarity as established in [Reference Dahlhaus14] for discrete-time locally stationary time series models. In [Reference Dahlhaus14], the authors consider a parametric representation of a sequence of non-stationary time-varying processes in either the time domain or the frequency domain, which has to satisfy certain regularity conditions.

In the following we briefly review the mathematical details of the aforementioned concepts, as well as the most important results. To this end, we define the total variation of a function g on [0, 1], denoted by V(g), as

\begin{align*}V(g)\;:\!=\;\sup\Bigg\{ \sum_{k=1}^m |g(x_k)-g(x_{k-1})|, 0\leq x_0 < \ldots < x_m \leq 1, m\in\mathbb{N} \Bigg\},\end{align*}

and for $\kappa>0$ we define

\begin{align*}\ell_\kappa(j)\;:\!=\;\begin{cases} 1,& |j|\leq1, \\[5pt] |j|\log^{1+\kappa}|j|, & |j|>1,\end{cases}\end{align*}

for all $j\in\mathbb{Z}$ . For further details on the following two definitions we refer to [Reference Dahlhaus14].

Definition 1. ([Reference Dahlhaus14, Assumption 1].) Let $\{X_{t,T}, t=1,\ldots,T\}_{T\in\mathbb{N}}$ be a sequence of stochastic processes. Then $X_{t,T}$ is called locally stationary in the time domain if there exists a representation

\begin{align*}X_{t,T}=\sum_{j=-\infty}^{\infty} a_{t,T,j}\varepsilon_{t-j}, \qquad T\in\mathbb{N}, \ t=1,\ldots,T,\end{align*}

where

1. (a) $\{\varepsilon_t,t\in\mathbb{Z}\}$ is an independent and identically distributed (i.i.d.) sequence with $\mathbb{E}[\varepsilon_t]=0$ and $Var(\varepsilon_t)=1$ ,

2. (b) for all $j\in\mathbb{Z}$ it holds that

   \begin{align*}\sup_{\substack{t=1,\ldots,T\\ T\in\mathbb{N}}}|a_{t,T,j}| \leq \frac{K}{\ell_\kappa(j)},\end{align*}
   where $\kappa,K>0$ are constants, and

3. (c) there exist functions $a_j(\!\cdot\!)\;:\;(0,1]\rightarrow\mathbb{R}$ , $j\in\mathbb{Z}$ , satisfying

   (1) \begin{align}\sup_{u\in(0,1]}|a_j(u)|\leq \frac{K}{\ell_\kappa(j)}, \qquad\sup_{j\in\mathbb{Z}}\sum_{t=1}^T \bigg|a_{t,T,j}-a_j\bigg(\frac{t}{T}\bigg)\bigg| \leq K , \quad \text{ and }\ V(a_j(\!\cdot\!)) \leq \frac{K}{\ell_\kappa(j)}\end{align}
   for some constant K.

Definition 2. ([Reference Dahlhaus14, p. 382].) Let $\{X_{t,T}\;:\; t=1,\ldots,T\}_{T\in\mathbb{N}}$ be a sequence of stochastic processes. Then $X_{t,T}$ is called locally stationary in the frequency domain with transfer functions $A_{t,T}^0\;:\;[\!-\!\pi,\pi]\rightarrow\mathbb{C}$ , $T\in\mathbb{N}$ , $t=1,\ldots,T$ , if it has the representation

\begin{align*}X_{t,T}=\int_{-\pi}^{\pi} e^{i\lambda t} A_{t,T}^0(\lambda) \xi(d\lambda) \qquad \text{for all } T\in\mathbb{N},\ t=1,\ldots,T\end{align*}

(with the integrals existing in $L^2$ ), where

1. (a) $\xi(\lambda)$ is a stochastic process on $[\!-\!\pi,\pi]$ with mean zero and orthogonal increments,

2. (b) there exist a constant K and a function $A\;:\;[0,1]\times[\!-\!\pi,\pi]\rightarrow\mathbb{C}$ which is continuous in the first component, satisfying $\overline{A(u,\lambda)}=A(u,-\lambda)$ and

   (2) \begin{align}\sup_{\substack{t=1,\ldots,T, \\ \lambda\in[\!-\!\pi,\pi]}}\bigg|A_{t,T}^0(\lambda)-A\bigg(\frac{t}{T},\lambda\bigg)\bigg| \leq \frac{K}{T}, \qquad T\in\mathbb{N}.\end{align}

The following two propositions give further insight into Definition 2 and the notion of local stationarity.

Proposition 1. Let $X_{t,T}$ be a locally stationary process in the frequency domain and $\{T_n\}_{n\in\mathbb{N}}\subset\mathbb{N}$ an increasing sequence. If $sT_n\in\{1,\ldots,T_n\}$ for some fixed $s\in[0,1]$ and all $n>n_0$ , $n_0\in\mathbb{N}$ , then it holds that

\begin{align*}A_{sT_n,T_n}^0(\!\cdot\!) \xrightarrow[n\rightarrow\infty]{L^2} A(s,\cdot).\end{align*}

For instance, the choice $T_n=2^n$ and $s=k/2^{n_0}$ for some $n_0\in\mathbb{N}$ and $k\in\{1,\ldots,T_{n_0}\}$ suits the conditions of Proposition 1.

Proposition 2. Let $X_{t,T}$ be a locally stationary process in the frequency domain with associated orthogonal increment process $\{\xi(\lambda),\lambda\in[\!-\!\pi,\pi]\}$ such that the sequence $\varepsilon_t = \int_{(-\pi,\pi]} \frac{1}{\sqrt{2\pi}} e^{i\lambda t} \xi(d\lambda)$ is i.i.d., and let $\{T_n\}_{n\in\mathbb{N}}\subset\mathbb{N}$ be an increasing sequence. If $sT_n\in\{1,\ldots,T_n\}$ for some fixed $s\in[0,1]$ and all $n>n_0$ , $n_0\in\mathbb{N}$ , then it holds that

\begin{align*}X_{sT_n,T_n}\overset{d}{\underset{n\rightarrow\infty}{\longrightarrow}}\int_{-\pi}^{\pi} A(s,\lambda) \xi(d\lambda).\end{align*}

Proof. First observe that every time series of the form $\int_{-\pi}^{\pi} e^{i\lambda t} A(\lambda) \xi(d\lambda)$ , $t\in\mathbb{Z}$ , where $\xi$ is an orthogonal increment process coming from an i.i.d. noise, is strictly stationary. Thus,

\begin{align*}\int_{-\pi}^{\pi} e^{i\lambda t_1} A_{sT_n,T_n}^0(\lambda) \xi(d\lambda)\overset{d}{=} \int_{-\pi}^{\pi} e^{i\lambda t_0} A_{sT_n,T_n}^0(\lambda) \xi(d\lambda)\end{align*}

for all $t_0,t_1\in\mathbb{Z}$ . In particular, for $t_1=sT_n$ , where $s\in[0,1]$ is such that $sT_n\in\{1,\ldots,T\}$ , and $t_0=0$ , we obtain

\begin{align*}X_{sT_n,T_n} =\int_{-\pi}^{\pi} e^{i\lambda sT_n} A_{sT_n,T_n}^0(\lambda) \xi(d\lambda) \overset{d}{=} \int_{-\pi}^{\pi} A_{sT_n,T_n}^0(\lambda) \xi(d\lambda).\end{align*}

The remainder follows from Proposition 1 and the continuity of the stochastic integral in mean square and thus in distribution with respect to the integrand.

Remark 2. A notable class of processes that are locally stationary in the time and frequency domains is that of time-varying AR(p) processes with continuous coefficient functions. For the mathematical details of this result we refer to [Reference Dahlhaus11, p. 147].

Among the variety of different concepts for local stationarity in the literature, we mention the results of [Reference Dahlhaus, Richter and Wu16, Reference Vogt44]. In [Reference Vogt44], the author considers a triangular array $X_{t,T}$ , $T\in\mathbb{N}$ , $t=1,\ldots,T$ , to be locally stationary if for each $u\in[0,1]$ there exists a strictly stationary process $\{X_t(u), t=1,\ldots,T\}$ such that almost surely

\begin{align*}\big| X_{t,T}-X_t(u) \big|\leq \bigg(\bigg| \frac{t}{T}-u \bigg|+\frac{1}{T}\bigg) U_{t,T}(u),\end{align*}

where $U_{t,T}(u)$ are positive random variables satisfying $\mathbb{E}[(U_{t,T}(u))^\rho]<\infty$ for some $\rho>0$ uniformly in u, t, and T. Time-varying AR(p) processes with continuous coefficient functions can also be embedded in this framework using arguments similar to those of [Reference Dahlhaus and Subba Rao17].

More recently, the authors of [Reference Dahlhaus, Richter and Wu16] developed a general theory for locally stationary processes based on stationary approximations. Similarly to [Reference Dahlhaus, Richter and Wu16, Reference Vogt44] assumes that there exists a strictly stationary process $\{X_t(u),t=1,\ldots,T\}$ such that for some $q,C>0$ ,

(3) \begin{align}\big\lVert X_t(u)-X_t(v) \big\rVert_{L^q}\leq C \big| u-v \big| \quad \text{and}\qquad \bigg\lVert X_{t,T}-X_t\bigg(\frac{t}{T}\bigg) \bigg\rVert_{L^q}\leq \frac{C}{T}\end{align}

uniformly in $t=1,\ldots,T$ and $u,v\in[0,1]$ . Based on these approximations, the authors establish asymptotic results as a law of large numbers and a central limit theorem, which in turn are used to derive asymptotic results for a maximum likelihood estimator (see [Reference Dahlhaus, Richter and Wu16, Section 5]). Again, time-varying AR(p) processes with continuous coefficient functions can be embedded in this framework. Recently, in [Reference Bardet, Doukhan and Wintenberger3], this work has been extended to models with infinite memory.

In view of the statistical results obtained from the approximations (3), a possible characterization of local stationarity in terms of similar approximations for continuous-time models will be the topic of future work.

2.2. Lévy processes and orthogonal random measures

In this section we lay the foundation for the definition of continuous-time locally stationary processes and briefly review Lévy processes and orthogonal random measures. We also cover basic results, including stochastic integration with respect to Lévy processes and orthogonal random measures. For further insight we refer to [Reference Applebaum1, Reference Sato39].

Theorem 1. Let $L=\{L(t),t\geq0\}$ be a real-valued Lévy process. Then L(1) is an infinitely divisible real-valued random variable with characteristic triplet $(\gamma,\Sigma,\nu)$ , where $\gamma \in \mathbb{R}$ , $\Sigma>0$ , and $\nu$ is a Lévy measure on $\mathbb{R}$ , i.e. $\nu(0)=0$ and $\int_{\mathbb{R}}(1\wedge\big| x \big|^2)\nu(dx)<\infty$ . The characteristic function of L(t) is given by

(4) \begin{align}\begin{aligned}\varphi_{L(t)}(z)&=\mathbb{E}\Big[e^{izL(t)}\Big]=\exp(t \Psi_{L}(z)),\\[5pt] \Psi_L(z)&=\Bigg( i\gamma z -\frac{\Sigma z^2}{2}+\int_{\mathbb{R} } \Big(e^{i z x}-1-i z x \mathbb{1}_{Z}(x) \Big)\nu(dx)\Bigg),\end{aligned}\end{align}

where $z \in \mathbb{R}$ and $Z=\{x \in \mathbb{R}, \big| x \big|\leq 1\}$ .

In the remainder we work with a two-sided Lévy process, i.e. $L(t)=L_1(t)\mathbb{1}_{\{t\geq0\}} - L_2(\!-\!t)\mathbb{1}_{\{t<0\}}$ , where $L_1$ and $L_2$ are independent copies of a one-sided Lévy process. Throughout this paper, it will be assumed that

(5) \begin{align} \mathbb{E}[L(1)]=\gamma + \int_{|x|> 1} x \nu(dx)=0 \text{ and } \mathbb{E}[L(1)^2]=\Sigma + \int_{x\in\mathbb{R}} x^2 \nu(dx)<\infty.\end{align}

Thus, the above assumptions on the Lévy process imply that $\int_\mathbb{R} x^2 \nu(dx)<\infty$ . Occasionally, we will write $\Sigma_L\;:\!=\;Var(L(1))=\Sigma+\int_{x\in\mathbb{R}} x^2 \nu(dx)$ .

If the Lévy process satisfies (5) and $f\;:\;\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}$ is a $\mathcal{B}(\mathbb{R}\times \mathbb{R}) - \mathcal{B}(\mathbb{R})$ -measurable function satisfying $f(t,\cdot)\in L^2(\mathbb{R})$ , then the integral $X(t)=\int_{\mathbb{R}} f(t,s) L(ds)$ , $t\in\mathbb{R}$ , exists in $L^2$ (see e.g. [Reference Marquardt and Stelzer28]).

In the proof of the above theorem, the standard theory of Fourier transforms on $L^2(\mathbb{R})$ (see e.g. [Reference Chandrasekharan10, Chapter 2] for an introduction) is used to show that

(6) \begin{align}\int_{-\infty}^\infty \varphi(\mu) \Phi_L(d\mu)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty \widehat{\varphi}(u) L(du)\end{align}

for any complex function $\varphi\in L^2(\mathbb{R})$ and its Fourier transform $\widehat{\varphi}$ , where

\begin{align*}\widehat{\varphi}(u)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty e^{-i\mu u} \varphi(\mu) d\mu.\end{align*}

Note that the above definition of the integral refers to the Plancherel extension theorem for the Fourier transform on $L^1(\mathbb{R})\cap L^2(\mathbb{R})$ to $L^2(\mathbb{R})$ . Hence, the corresponding integral is understood symbolically as the corresponding limit in $L^2(\mathbb{R})$ . Similarly, we define the inverse Fourier transform g of a function $\widehat{g}\in L^2(\mathbb{R})$ symbolically as

\begin{align*}g(\mu)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty e^{i\mu u} \widehat{g}(u) du.\end{align*}

We also recall that for two complex functions $f,g\in L^2(\mathbb{R})$ and their Fourier transforms $\widehat{f}, \widehat{g}$ , it follows that $\hat{f}, \hat{g}\in L^2(\mathbb{R})$ and, by [Reference Rudin37, p. 189],

\begin{align*}\int_{-\infty}^\infty f(\mu)\overline{g(\mu)} d\mu = \int_{-\infty}^\infty \widehat{f}(u)\overline{\widehat{g}(u)} du.\end{align*}

4.1. Locally stationary CAR(1) processes

The simplest Lévy-driven CARMA process is the Lévy-driven CAR(1) or Ornstein–Uhlenbeck-type process.

For a constant coefficient $a>0$ , a CAR(1) process is the stationary solution to the stochastic differential equation $dY(t)=-aY(t)dt+L(dt)$ , which can be expressed as

\begin{align*}Y(t)=\int_{-\infty}^t e^{-a (t-u)} L(du).\end{align*}

We replace the constant a by a time-varying function a(t) and arrive at a so-called time-varying CAR(1) process, which is given by

\begin{align*}Y(t)=\int_{-\infty}^t e^{-\int_u^t a(s) ds} L(du).\end{align*}

Additional rescaling results in a sequence of time-varying CAR(1) processes that could be locally stationary. We consider the sequence of stochastic processes $\{Y_N(t), t\in\mathbb{R}\}_{N\in\mathbb{N}}$ defined by

(8) \begin{align} Y_N(t) &= \int_{-\infty}^{Nt} e^{-\int_u^{Nt} a(\frac{s}{N}) ds} L(du),\end{align}

where $a\;:\;\mathbb{R}\rightarrow\mathbb{R}$ is a continuous coefficient function such that

\begin{align*} u\mapsto e^{-\int_u^{Nt} a(\frac{s}{N}) ds}\end{align*}

is square-integrable for all $t\in\mathbb{R}$ , $N\in\mathbb{N}$ , and L is a two-sided Lévy process. Recall that the Lévy process satisfies (5). In view of Definition 3, we obtain from (8) that

(9) \begin{align}\begin{aligned}g_N^0(Nt,Nt-u) &= \mathbb{1}_{\{Nt-u\geq 0\}} e^{-\int_u^{Nt} a(\frac{s}{N}) ds} &&\!\!\!= \mathbb{1}_{\{Nt-u\geq 0\}} e^{-\int_{-(Nt-u)}^0 a(\frac{s}{N}+t) ds} \quad \text{ and} \\[5pt] A_N^0(Nt,\mu) &= \int_\mathbb{R} e^{-i\mu u} g_N^0(Nt,u) du &&\!\!\!= \int_\mathbb{R} e^{-i\mu v} \mathbb{1}_{\{v\geq 0\}} e^{-\int_{-v}^0 a(\frac{s}{N}+t) ds} dv.\end{aligned}\end{align}

Proposition 6. Let $\{Y_N(t), t\in\mathbb{R}\}_{N\in\mathbb{N}}$ be a sequence of time-varying CAR(1) processes as defined in (8). If

1. (C1) $a(\!\cdot\!)$ is continuous and

2. (C2) for every $T\in\mathbb{R}^+$ there exists $\varepsilon_T>0$ such that $a(s)\geq \varepsilon_T$ for all $s\leq T$ ,

then $Y_N(t)$ is well-defined and locally stationary, with the limiting kernel g and limiting transfer function given by

\begin{align*}g(t,u) =\mathbb{1}_{\{u\geq 0\}}e^{-a(t)u}\quad \textit{ and}\qquad A(t,\mu)=\int_\mathbb{R} e^{-i\mu u} \mathbb{1}_{\{u\geq 0\}}e^{-a(t)u} du.\end{align*}

Proof. From (C2) we readily obtain that $Y_N(t)$ is well-defined. For all $t\in\mathbb{R}$ it holds that

\begin{align*}\Big\lVert g_N^0(Nt,\cdot)-g(t,\cdot) \Big\rVert_{L^2}^2 &=\Big\lVert g_N^0(Nt,Nt-\cdot)-g(t,Nt-\cdot) \Big\rVert_{L^2}^2 \\[5pt] &=\int_\mathbb{R}\! \Bigg| \mathbb{1}_{\{Nt-u\geq 0\}} e^{-\int_{-(Nt-u)}^0 a(\frac{s}{N}+t) ds} - \mathbb{1}_{\{Nt-u\geq 0\}}e^{-a(t)(Nt-u)} \Bigg|^2 du \\[5pt] &= \int_\mathbb{R} \mathbb{1}_{\{u\leq 0\}} \Bigg| e^{-\int_u^0 a(\frac{s}{N}+t) ds} - e^{a(t)u} \Bigg|^2 du\underset{N\rightarrow\infty}{\longrightarrow}0,\end{align*}

using the dominated convergence theorem. For the inner integral, the continuity of a on a compact set is sufficient for an application of the dominated convergence theorem. As majorant for the outer integral we consider $u\mapsto4\mathbb{1}_{\{u\leq 0\}} e^{2\varepsilon_t u}\in L^2$ . The required $L^2$ -continuity of the limiting kernel g can be obtained similarly, using Lemma 1.

4.2. Time-varying CARMA(p, q) processes and time-varying state space models

Consider $p,q\in\mathbb{N}$ , where $p>q$ . The formal differential equation for a time-varying Lévy-driven CARMA(p, q) process is given by $p(t,D)Y(t) = q(t,D)DL(t)$ , i.e.

\begin{align*}&D^p Y(t) + a_1(t) D^{p-1}Y(t) + \ldots +a_p(t) Y(t) \\[5pt] &= b_0(t) DL(t) + b_1(t) D^2L(t) +\ldots +b_q(t) D^{q+1}L(t),\end{align*}

where D denotes the differential operator with respect to time and L(t) is a two-sided Lévy process satisfying (5). For continuous functions $a_i(t),b_i(t)$ , $i=1,\ldots,p$ , where $b_i(t)=0$ for all $i>q$ , the polynomials

(10) \begin{align}\begin{aligned} p(t,z) &= z^p+a_1(t)z^{p-1}+\ldots+a_{p-1}(t)z+a_p(t) \quad \text{ and} \\[5pt] q(t,z) &= b_0(t)+b_1(t)z+\ldots+b_{q-1}(t)z^{q-1}+b_q(t)z^q\end{aligned} \end{align}

are called autoregressive (AR) and moving average (MA) polynomials. For a rigorous definition we interpret the differential equations to be equivalent to the state space representation

(11) \begin{align}\begin{aligned}Y(t) &= \mathcal{B}(t)' \mathcal{X}(t), \quad \text{ and}\\[5pt] d\mathcal{X}(t) &= \mathcal{A}(t) \mathcal{X}(t) dt + \mathcal{C} L(dt), \quad t\in\mathbb{R},\end{aligned}\end{align}

with

\begin{align*}\mathcal{A}(t)&=\left( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c}0 & 1 & \ldots & 0 \\[5pt] \vdots & \; & \ddots & \vdots \\[5pt] 0 & \; & \; & 1 \\[5pt] -a_p(t) & -a_{p-1}(t) & \ldots & -a_1(t) \\[5pt] \end{array}\right)\in M_{p\times p}(\mathcal{R}[t]) \quad \text{ and}\\[5pt] \mathcal{B}(t)&=\left( \begin{array}{c}b_0(t) \\[5pt] b_1(t)\\[5pt] \vdots \\[5pt] b_{p-1}(t)\\[5pt] \end{array}\right)\in M_{p\times 1}(\mathcal{R}[t]), \qquad\mathcal{C}=\left( \begin{array}{c}0 \\[5pt] \vdots\\[5pt] 0 \\[5pt] 1\\[5pt] \end{array}\right)\in M_{p\times 1}(\mathbb{R}),\end{align*}

where $\mathcal{R}[t]$ denotes the ring of continuous functions in t from $\mathbb{R}$ to $\mathbb{R}$ .

It is obvious that (11) has a unique solution when one fixes the value $X(t_0)$ at some point $t_0\in\mathbb{R}$ . For a Brownian motion as driving noise, such equations were investigated in [Reference Surulescu43, Section 2.1.1].

Provided the integrals exist in $L^2$ , it can be shown that a solution is given by

(12) \begin{align}\mathcal{X}(t) = \int_{-\infty}^t \Psi(t,s) \mathcal{C} L(ds) \quad \text{and} \quad Y(t) = \mathcal{B}(t)' \int_{-\infty}^t \Psi(t,s) \mathcal{C} L(ds),\end{align}

where $\Psi(t,t_0)$ is the unique matrix solution of the homogeneous initial value problem (IVP) $\frac{d}{dt}\Psi(t,t_0)=\mathcal{A}(t)\Psi(t,t_0)$ , where $\Psi(t_0,t_0)=\textbf{1}_p$ for all $t>t_0$ (see [Reference Brockett7, Sections 3 and 4]). The transition matrix $\Psi$ satisfies $\Psi(t,t_0)=\Psi(t,u)\Psi(u,t_0)$ for all $t>u>t_0$ (see [Reference Brockett7, Section 4, Theorem 2]). In particular, the integrals in (12) are well-defined (see Section 2.2) if there exist $\gamma,\lambda>0$ such that

\begin{align*}\big\lVert \Psi(t,t_0) \big\rVert \leq \gamma e^{-\lambda(t-t_0)}\qquad \text{ for all } t,t_0 \text{ with } t\geq t_0.\end{align*}

This condition corresponds to uniform exponential stability of the state space model in (11) and will be explained in more detail in Section 4.3.

The usual integral representation of stationary causal CARMA processes motivates the following definition.

For some initial time $t_0\in\mathbb{R}$ the process satisfies the following relation (see [Reference Surulescu43, Section 2.1.1]):

(13) \begin{align}\mathcal{X}(t) = \Psi(t,t_0)\Bigg(\mathcal{X}(t_0)+\int_{t_0}^t \Psi(u,t_0)^{-1} \mathcal{C} L(du)\Bigg).\end{align}

From [Reference Baake and Schlägel2, Remark 2] it follows that if for all $t,t_0\in\mathbb{R}$ and $t>t_0$

(14) \begin{align}\mathcal{A}(t)\int_{t_0}^t \mathcal{A}(s)ds = \int_{t_0}^t \mathcal{A}(s)ds \mathcal{A}(t),\end{align}

then $\Psi(t,t_0)=e^{\int_{t_0}^t \mathcal{A}(s)ds}$ .

If the commutativity assumption (14) holds, the equations (12) and (13) simplify to

\begin{align*}\mathcal{X}(t) &= e^{\int_{t_0}^t \mathcal{A}(s)ds}\mathcal{X}(t_0)+\int_{t_0}^t e^{\int_u^t \mathcal{A}(s)ds} \mathcal{C} L(du)\!\!\! &&= \int_{-\infty}^t e^{\int_u^t \mathcal{A}(s)ds} \mathcal{C} L(du) \quad \text{ and} \\[9pt] Y(t) &= \mathcal{B}(t)' e^{\int_{t_0}^t \mathcal{A}(s)ds}\mathcal{X}(t_0)+\int_{t_0}^t \mathcal{B}(t)' e^{\int_u^t \mathcal{A}(s)ds} \mathcal{C} L(du)\!\!\!&&= \int_{-\infty}^t \mathcal{B}(t)' e^{\int_u^t \mathcal{A}(s)ds} \mathcal{C} L(du)\end{align*}

for $t,t_0\in\mathbb{R}$ , where $t>t_0$ .

If the assumption (14) does not hold, $\Psi(t,t_0)$ can be expressed by the Peano–Baker series (see [Reference Baake and Schlägel2, Section 2])

\begin{align*}\Psi(t,t_0) &= \textbf{1}_p + \int_{t_0}^t \mathcal{A}(\tau_1) d\tau_1 + \int_{t_0}^t \mathcal{A}(\tau_1) \int_{t_0}^{\tau_1} \mathcal{A}(\tau_2) d\tau_2 d\tau_1 + \ldots = \sum_{n=0}^\infty \mathcal{I}_n(t),\end{align*}

where $\mathcal{I}_0(t)= \textbf{1}_p$ and $\mathcal{I}_n(t)= \int_{t_0}^t \mathcal{A}(\tau_1) \int_{t_0}^{\tau_1} \mathcal{A}(\tau_2) \cdots \int_{t_0}^{\tau_{n-1}} \mathcal{A}(\tau_n) d\tau_n \ldots d\tau_2 d\tau_1$ .

The previous remark shows that, when considering time-varying CARMA(p, q) processes, it is in general not possible to describe the solution of the state space equations explicitly in the form of a matrix exponential. Instead one has to use the Peano–Baker series.

In [Reference Schlemm and Stelzer41, Corollary 3.4] it is proved that in the time-invariant case, the class of CARMA processes is equivalent to the class of continuous-time state space models. This motivates us to look at time-varying state space processes. We consider the observation and state equations

(15) \begin{align}\begin{aligned}Y(t) &= B(t)' X(t) \quad \text{ and} \\[5pt] dX(t) &= A(t)X(t)dt + C(t) L(dt),\end{aligned}\end{align}

where $t\in\mathbb{R}$ , $A(t)\in M_{p\times p}(\mathcal{R}[t])$ and $B(t),C(t)\in M_{p\times1}(\mathcal{R}[t])$ are arbitrary continuous coefficient functions, and L is a two-sided Lévy process satisfying (5).

Now, the representation of a time-varying CARMA process as given in (12) can be adapted to (general) state space processes. Provided the integrals exist in $L^2$ , it can be shown that a solution of (15) is given by

(16) \begin{align}X(t) = \int_{-\infty}^t \Psi(t,u) C(u) L(du) \quad \text{and} \quad Y(t) = B(t) \int_{-\infty}^t \Psi(t,u) C(u) L(du),\end{align}

where $\Psi(t,t_0)$ is the unique matrix solution of the IVP $\frac{d}{dt}\Psi(t,t_0)=A(t)\Psi(t,t_0)$ , $\Psi(t_0,t_0)=\textbf{1}_p$ , for $t>t_0$ . In particular, in the representation of (16), the integrals are well-defined if $C(\!\cdot\!)\in L^2(\mathbb{R})$ and there exist $\gamma,\lambda>0$ such that

\begin{align*}\big\lVert \Psi(t,t_0) \big\rVert \leq \gamma e^{-\lambda(t-t_0)}\qquad \text{for all }t\geq t_0 \text{ (uniform exponential stability)}.\end{align*}

For some initial time $t_0\in\mathbb{R}$ , the process satisfies the relation

(17) \begin{align}X(t) = \Psi(t,t_0)\Bigg(X(t_0)+\int_{t_0}^t \Psi(u,t_0)^{-1} C(u) L(du)\Bigg).\end{align}

Finally, we make the following definition.

The natural question arises of whether all time-varying state space processes are tvCARMA processes, as in the time-invariant case. A comprehensive investigation of this question seems to be beyond the scope of this work. Below we present a result indicating that this is probably not the case in general (and definitely not if we allow the coefficient functions to have a discontinuity). Moreover, we give sufficient conditions for a positive answer.

Proof. Consider a two-dimensional time-varying state space model as defined in (15) with a structural break at $t=1$ . As coefficient functions we consider the step functions

(18) \begin{align}B(t)=\begin{cases} B_1 &\text{ if } t\leq1, \\[5pt] B_2 & \text{ if } t>1, \end{cases} \qquad A(t)=\begin{cases} A_1 &\text{ if } t\leq1, \\[5pt] A_2 & \text{ if } t>1, \end{cases} \quad \text{ and} \quad C(t)=\begin{cases} C_1 &\text{ if } t\leq1, \\[5pt] C_2 & \text{ if } t>1, \end{cases}\end{align}

which satisfy the uniform exponential stability assumption for the solution of (15).

We assume that the system is in the form of a CARMA process for $t\leq1$ and assume (for contradiction) that there exists an equivalent CARMA model as defined in (11) for all $t\in\mathbb{R}$ . Then the CARMA model shows the same structural resemblance as the corresponding state space model. In the following we denote the coefficients of the CARMA model by $\mathcal{B}(t)$ , $\mathcal{A}(t)$ , and $\mathcal{C}(t)$ . Using the same notation as in (18), we obtain

(19) \begin{align}&\mathcal{B}_1 = B_1, &&\mathcal{A}_1 = A_1, &&\mathcal{C}_1 = C_1, \nonumber \\&\mathcal{B}_2 = \begin{pmatrix} \ast \\ \ast \end{pmatrix}, &&\mathcal{A}_2 = \begin{pmatrix} 0\;\;\;\; & 1 \\ \ast\;\;\;\; & \ast \end{pmatrix},\;\;\;\;\;\;\;\text{ and}&&\mathcal{C}_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}.\end{align}

Since the structural break divides the model into two separate linear models, the CARMA representations $(\mathcal{B}_1,\mathcal{A}_1,\mathcal{C}_1)$ and $(\mathcal{B}_2,\mathcal{A}_2,\mathcal{C}_2)$ are unique. From the proof of [Reference Schlemm and Stelzer41, Theorem 3.3] we obtain that $\mathcal{B}_2 e^{\mathcal{A}_2 (t-u)} \mathcal{C}_2 = B_2 e^{A_2 (t-u)} C_2$ for all $t>1$ and $u\in(1,t]$ .

On the one hand, we have

(20) \begin{align}\begin{aligned}Y(t) &= \mathbb{1}_{\{t\leq1\}} \Bigg( \int_{-\infty}^t B(t)' \Psi_A(t,u) C(u) L(du) \Bigg) \\[5pt] &\quad+ \mathbb{1}_{\{t>1\}} \Bigg( B(t)' \Psi_A(t,1) X(1) + B(t)' \Psi_A(t,1) \int_1^t \Psi_A(u,1)^{-1} C(u) L(du) \Bigg) \\[5pt] &= \mathbb{1}_{\{t\leq1\}} \Bigg( \int_{-\infty}^t {B}'_{\!\!1} e^{A_1(t-u)} C_1 L(du) \Bigg)\\[5pt] &\quad+ \mathbb{1}_{\{t>1\}} \Bigg( {B}'_{\!\!2} e^{A_2(t-1)} X(1) + \int_1^t {B}'_{\!\!2} e^{A_2(t-u)} C_2 L(du) \Bigg),\end{aligned}\end{align}

where $\Psi_A(s,s_0)$ denotes the solution of the aforementioned IVP with respect to A. On the other hand, Y(t) can be written as

(21) \begin{align}\begin{aligned}Y(t)&= \mathbb{1}_{\{t\leq1\}} \Bigg( \int_{-\infty}^t \mathcal{B}_1' e^{\mathcal{A}_1(t-u)} \mathcal{C}_1 L(du) \Bigg) \\[5pt] &\quad+\mathbb{1}_{\{t>1\}} \Bigg( \mathcal{B}'_{\!\!2} e^{\mathcal{A}_2(t-1)} \mathcal{X}(1) + \int_1^t \mathcal{B}'_{\!\!2} e^{\mathcal{A}_2(t-u)} \mathcal{C}_2 L(du) \Bigg).\end{aligned}\end{align}

From (19) it follows that

\begin{align*}X(1) &= \int_{-\infty}^1 e^{A_1(1-u)} C_1 L(du) =\int_{-\infty}^1 e^{\mathcal{A}_1(1-u)} \mathcal{C}_1 L(du)=\mathcal{X}(1).\end{align*}

Thus, combining (20) and (21), using (19) and the independent increments of the Lévy process, the equality ${B}'_{\!\!2} e^{A_2(t-1)} X(1)=\mathcal{B}'_{\!\!2} e^{\mathcal{A}_2(t-1)} X(1)$ has to hold almost surely for all $t>1$ . Therefore, for almost all x in the support of X(1) we obtain

(22) \begin{align}{B}'_{\!\!2} e^{A_2(t-1)} x=\mathcal{B}'_{\!\!2} e^{\mathcal{A}_2(t-1)} x.\end{align}

In the sequel we give a particular Lévy process and coefficient functions that lead to a contradiction in (22).

Assume that the Lévy process is a Brownian motion. Thus, it has the triplet $(0,\Sigma,0)$ for some $\Sigma>0$ . From [Reference Sato40] it follows that X(t) is a Lévy process with triplet $(0,\Sigma_X^t,0)$ , where

\begin{align*} \Sigma_X^1= \int_{-\infty}^1 e^{A_1(1-u)} C_1 \Sigma C'_{\!\!1} e^{A'_{\!\!1}(1-u)} du = \Sigma \int_0^\infty e^{A_1 u} C_1 C'_{\!\!1} e^{A'_{\!\!1} u} du \in M_{2\times 2}(\mathbb{R}).\end{align*}

The regularity of $\Sigma_X^1$ can be shown by investigating $Im(\Sigma_X^1)$ , where $Im(D)=\{Dx\;:\; x\in\mathbb{R}^d\}$ denotes the image of a matrix $D\in M_{d\times d}(\mathbb{R})$ . Using [Reference Bernstein6, Lemma 12.6.2] (see also [Reference Schlemm and Stelzer41, p. 54]), we obtain

\begin{align*}Im\Bigg(\int_0^\infty e^{A_1 u} C_1 C'_{\!\!1} e^{A'_{\!\!1} u} du\Bigg) = Im\bigg(\bigg[C_1\ \ A_1 C_1\ \ \cdots\ \ A_1^{p-1}C_1\bigg]\bigg).\end{align*}

Therefore, in our setting, it is sufficient to find $A_1,C_1$ such that $[C_1 \ A_1 C_1]$ is regular, which also implies that $\Sigma_X^1$ is positive definite. Then X(1) has characteristic function

\begin{align*} \mathbb{E}\Big[e^{i\langle z,X(1) \rangle }\Big] = e^{-\frac{1}{2}\langle z,\Sigma_X^1 z \rangle},\end{align*}

which corresponds to a two-dimensional $N(0,\Sigma_X^1)$ -distributed random variable having positive density for all values $x\in\mathbb{R}^2$ . To contradict (22), it is enough to show that for some $t>1$ ,

\begin{align*}{B}'_{\!\!2} e^{A_2(t-1)} x \neq \mathcal{B}'_{\!\!2} e^{\mathcal{A}_2(t-1)} x \qquad \text{ for all } x\in I \text{, where } I\subset\mathbb{R}^2 \text{ with } \lambda(I)>0.\end{align*}

We define

\begin{align*} &B_1 = \begin{pmatrix} 1 \\[5pt] 2 \end{pmatrix},\ B_2 = \begin{pmatrix} 1 \\[5pt] 1 \end{pmatrix},\ \mathcal{B}_2 = \begin{pmatrix} 5 \\[5pt] 2 \end{pmatrix}, \quad A_1 = \begin{pmatrix} 0\;\;\;\; & 1 \\[5pt] 1\;\;\;\; &1 \end{pmatrix}, \ A_2 = \begin{pmatrix} -2\;\;\;\; & 0 \\[5pt] 0\;\;\;\; & -3 \end{pmatrix},\\[5pt] &\mathcal{A}_2 = \begin{pmatrix} 0\;\;\;\; & 1 \\[5pt] -6\;\;\;\; & -5 \end{pmatrix},\ C_1 = \begin{pmatrix} 0 \\[5pt] 1 \end{pmatrix},\ C_2 = \begin{pmatrix} 1 \\[5pt] 1 \end{pmatrix},\ \text{ and } \mathcal{C}_2 = \begin{pmatrix} 0 \\[5pt] 1 \end{pmatrix}.\end{align*}

From this we obtain that the CARMA model has the same transfer function as the state space model, since

\begin{align*} {B}'_{\!\!2} (z\textbf{1}_2-A_2)^{-1} C_2 = \frac{2z+5}{z^2+5z+6}=\mathcal{B}'_{\!\!2} (z-\textbf{1}_2\mathcal{A}_2)^{-1} \mathcal{C}_2.\end{align*}

Moreover,

\begin{align*} [C_1\ \ A_1 C_1]=\begin{pmatrix} 0\;\;\;\;\; & 1 \\[5pt] 1\;\;\;\;\; & 1\end{pmatrix}\end{align*}

is regular. Given a vector $x=(x_1, x_2)'$ , it remains to investigate $\mathcal{B}'_{\!\!2} e^{\mathcal{A}_2(t-1)} x - {B}'_{\!\!2} e^{A_2(t-1)} x$ . For a matrix $D\in\mathbb{C}^{2\times 2}$ with eigenvalues $\sigma(A)=\{\mu,\lambda\}$ , [Reference Bernstein6, Proposition 11.3.2] gives that

\begin{align*}e^D=\begin{cases} e^\lambda ((1-\lambda)\textbf{1}_2+D) & \text{if } \mu=\lambda, \\[5pt] \frac{\mu e^\lambda - \lambda e^\mu}{\mu-\lambda} \textbf{1}_2 + \frac{e^\mu - e^\lambda}{\mu-\lambda} D & \text{if } \mu\neq\lambda. \end{cases}\end{align*}

Since $\sigma(\mathcal{A}_2(t-1))=\{-2(t-1),-3(t-1)\}$ , we obtain

\begin{align*}&\mathcal{B}'_{\!\!2} e^{\mathcal{A}_2(t-1)} x - {B}'_{\!\!2} e^{A_2(t-1)} x \\[5pt] &= \begin{pmatrix} 5\;\;\;\;\; & 2 \end{pmatrix}\Bigg(\frac{-3(t-1) e^{-2(t-1)} - (\!-\!2)(t-1) e^{-3(t-1)}}{-3(t-1)-(\!-\!2)(t-1)} \begin{pmatrix} 1\;\;\;\;\; & 0 \\[5pt] 0\;\;\;\;\; & 1 \end{pmatrix} \\[5pt] &\qquad\qquad\qquad+ \frac{e^{-3(t-1)} - e^{-2(t-1)}}{-3(t-1)-(\!-\!2)(t-1)} \begin{pmatrix} 0\;\;\;\;\; & 1 \\[5pt] -6\;\;\;\;\; & -5 \end{pmatrix} (t-1) \Bigg)\begin{pmatrix} x_1 \\[5pt] x_2 \end{pmatrix} \\[5pt] &\qquad\qquad\qquad- \begin{pmatrix} 1\;\;\;\;\; & 1 \end{pmatrix}\Bigg(\begin{pmatrix} e^{-2(t-1)}\;\;\;\;\; & 0 \\[5pt] 0\;\;\;\;\; & e^{-3(t-1)} \end{pmatrix}\Bigg)\begin{pmatrix} x_1 \\[5pt] x_2 \end{pmatrix} \\[5pt] &= 2x_1 \Big( e^{-2(t-1)} + e^{-3(t-1)}\Big) +x_2 \Big( e^{-2(t-1)} \Big) >0\end{align*}

for all $x\in I=\{x\in\mathbb{R}^2 \;:\; x_1>0,x_2>0\}$ and $t>1$ .

Under more rigorous conditions on the coefficient functions, the concept of controllability from linear system theory allows for a characterization for special canonical forms, which occur in the state space representation of CARMA processes ( $\mathcal{A}$ is in companion matrix form). The following results summarize the key aspects of this characterization, which is based mainly on [Reference Silverman42], but also on [Reference Benmahammed5, Reference Ramar and Ramaswami32, Reference Ramaswami and Ramar33].

Definition 9. ([Reference Rugh38, Chapters 9 and 10].) Let Y(t) be a state space model as defined in (15), where A(t) is $(p-1)$ times continuously differentiable and C(t) is p times continuously differentiable. We define the controllability matrix $W_p(t)$ as

\begin{align*}W_p(t)&=[K_0(t)\ K_1(t)\ \cdots \ K_{p-1}(t)], \quad \text{ where}\\[5pt] K_0(t)&=C(t), \quad K_{i+1}(t) = - A(t) K_i(t) + \frac{d}{dt}K_i(t), \quad i=1,\ldots,p-2.\end{align*}

Then the state space process X(t) is called

1. (a) controllable on $[t_0,t_1]$ , $t_0<t_1$ , if there exists $t\in[t_0,t_1]$ with $Rank(W_p(t))=p$ , and

2. (b) instantaneously controllable if $Rank(W_p(t))=p$ for all $t\in\mathbb{R}$ .

Proposition 8. ([Reference Silverman42, Theorem 1.].) Consider a state space process satisfying (15) such that A is $(p-1)$ times continuously differentiable and C is p times continuously differentiable. Then it is equivalent to a CARMA process satisfying (11) if and only if it is instantaneously controllable. Equivalence means that there exists a regular matrix $T(t)\in M_{n\times n}(\mathcal{R}[t])$ which is continuously differentiable and satisfies

\begin{align*}\mathcal{X}(t) = T(t) X(t)\end{align*}

almost surely. The relationship between the two systems is given by $T(t) = \mathcal{W}_p(t) W_p(t)^{-1}$ , $\mathcal{A}(t) = \Big( T(t) A(t) + \frac{d}{dt}T(t) \Big) T(t)^{-1}$ , and $\mathcal{C} = T(t) C(t)$ , where $\mathcal{W}_p(t)$ and $W_p(t)$ are the controllability matrices of the state space model and the CARMA process.

Proof. Any time-varying CARMA(p, q) process is obviously also a time-varying state space process. For the converse, let Y(t) be a time-varying state space process defined by (15) which is instantaneously controllable with controllability matrix $W_p(t)$ . Then, by Proposition 8, the state system $dX(t) = A(t)X(t)dt + C(t) L(dt)$ is equivalent to the CARMA system $d\mathcal{X}(t) = \mathcal{A}(t)\mathcal{X}(t)dt+\mathcal{C} L(dt)$ with

\begin{align*}\mathcal{X}(t) = T(t) X(t), \quad \mathcal{C} = T(t) C(t), \quad \text{and} \quad \mathcal{A}(t) = \Bigg( T(t) A(t) + \frac{d}{dt}T(t) \Bigg) T(t)^{-1},\end{align*}

where $T(t) = \mathcal{W}_p(t) W_p(t)^{-1}$ is regular. Thus,

\begin{align*}Y(t)&= B(t)' X(t) = B(t)' T(t)^{-1} \mathcal{X}(t) = \mathcal{B}(t)' \mathcal{X}(t)\quad \text{and}\\[5pt] d\mathcal{X}(t) &= \mathcal{A}(t)\mathcal{X}(t)dt+\mathcal{C} L(dt),\end{align*}

which is a representation for Y(t) as a time-varying CARMA(p, q) process in (11).

4.3. Locally stationary linear state space models—Peano–Baker series

We investigate sufficient conditions for sequences of time-varying state space processes, which obviously also include sequences of time-varying CARMA processes, to be locally stationary.

Let $\{Y_N(t), t\in\mathbb{R}\}_{N\in\mathbb{N}}$ be a sequence of time-varying linear state space processes defined by

\begin{align*}Y_N(t) &= B(t)' X_N(Nt)\quad \text{and} \\[5pt] X_N(Nt) &= \Psi_N(Nt,0) \int_{-\infty}^{Nt} \Psi_N(u,0)^{-1} C\bigg(\frac{u}{N}\bigg) L(du),\end{align*}

where $\Psi_N(s,s_0)$ is the solution of the matrix differential equation

\begin{align*}\Psi_N(s_0,s_0) &= \textbf{1}_p, \\[5pt] \frac{d}{ds}\Psi_N(s,s_0) &= A\bigg(\frac{s}{N}\bigg) \Psi_N(s,s_0), \quad \text{for all } s,s_0\in\mathbb{R} \text{ with } s>s_0,\end{align*}

which can be expressed as follows (see [Reference Baake and Schlägel2, Section 2]):

\begin{align*}\Psi_N(s,s_0) = \textbf{1}_p + \int_{s_0}^s A\bigg(\frac{\tau_1}{N}\bigg) d\tau_1 + \int_{s_0}^s A\bigg(\frac{\tau_1}{N}\bigg) \int_{s_0}^{\tau_1} A\bigg(\frac{\tau_2}{N}\bigg) d\tau_2 d\tau_1 + \ldots.\end{align*}

The substitution $s\mapsto s+Nt$ in (9) is necessary to achieve a dependence of the kernel function $g_N^0(t,\cdot)$ on $Nt-u$ . Therefore, we define $\widetilde{\Psi}_{N,t}(0,-(Nt-u))$ for a fixed point $t\in\mathbb{R}$ as the solution of the matrix differential equation

\begin{align*}\widetilde{\Psi}_{N,t}(s_0,s_0) &= \textbf{1}_p, \\[5pt] \frac{d}{ds}\widetilde{\Psi}_{N,t}(s,s_0) &= A\bigg(\frac{s}{N}+t\bigg) \widetilde{\Psi}_{N,t}(s,s_0), \quad \text{for all } s,s_0\in\mathbb{R} \text{ with } s>s_0,\end{align*}

which can again be expressed as

\begin{align*}\widetilde{\Psi}_{N,t}(s,s_0)= \textbf{1}_p + \int_{s_0}^s A\bigg(\frac{\tau_1}{N}+t\bigg) d\tau_1 + \int_{s_0}^s A\bigg(\frac{\tau_1}{N}+t\bigg) \int_{s_0}^{\tau_1} A\bigg(\frac{\tau_2}{N}+t\bigg) d\tau_2 d\tau_1 + \ldots.\end{align*}

From [Reference Brockett7, Theorem 4.2] we obtain

\begin{align*} \Psi_N(Nt,0)\Psi_N(u,0)^{-1} = \Psi_N(Nt,0)\Psi_N(0,u) = \Psi_N(Nt,u).\end{align*}

Since

\begin{align*}\Psi_N(Nt,u) &= \textbf{1}_p + \int_{u}^{Nt} A\bigg(\frac{\tau_1}{N}\bigg) d\tau_1 + \int_{u}^{Nt} A\bigg(\frac{\tau_1}{N}\bigg) \int_{u}^{\tau_1} A\bigg(\frac{\tau_2}{N}\bigg) d\tau_2 d\tau_1 + \ldots \\[5pt] &= \textbf{1}_p + \int_{u-Nt}^0 A\bigg(\frac{\tau_1}{N}+t\bigg) d\tau_1 + \int_{u-Nt}^0 A\bigg(\frac{\tau_1}{N}+t\bigg) \int_{u-Nt}^{\tau_1} A\bigg(\frac{\tau_2}{N}+t\bigg) d\tau_2 d\tau_1 + \ldots \\[5pt] &= \widetilde{\Psi}_{N,t}(0,-(Nt-u)),\end{align*}

we neglect the superscript tilde and define a sequence of time-varying linear state space processes as follows.

Definition 10. A sequence of time-varying linear state space processes $\{Y_N(t), t\in\mathbb{R}\}_{N\in\mathbb{N}}$ is defined as

(23) \begin{align}Y_N(t) = \int_\mathbb{R} \mathbb{1}_{\{Nt-u\geq 0\}} B(t)' \Psi_{N,t}^0(0,-(Nt-u)) C\bigg(\frac{u}{N}\bigg) L(du) \end{align}

with (limiting) kernel function (in view of Definition 5) given by

\begin{align*}g_N^0(Nt,Nt-u) &= \mathbb{1}_{\{Nt-u\geq 0\}} B(t)' \Psi_{N,t}^0(0,-(Nt-u)) C\bigg(\frac{-(Nt-u)}{N}+t\bigg) \quad\text{and}\\[5pt] g(t,Nt-u) &= \mathbb{1}_{\{Nt-u\geq 0\}} B(t)' \Psi_t(0,-(Nt-u)) C(t),\end{align*}

where $\Psi_{N,t}^0(0,-(Nt-u))$ and $\Psi_t(0,-(Nt-u))$ are the solutions of the matrix differential equations

(24) \begin{align}\begin{aligned}\Psi_{N,t}^0(s_0,s_0) &= \textbf{1}_p, \quad &&\frac{d}{ds} \Psi_{N,t}^0(s,s_0) = A\bigg(\frac{s}{N}+t\bigg) \Psi_{N,t}^0(s,s_0),\\[5pt] \Psi_t(s_0,s_0)&= \textbf{1}_p, \quad \text{and} \quad && \frac{d}{ds} \Psi_t(s,s_0)= A(t) \Psi_t(s,s_0) \end{aligned}\end{align}

for $s>s_0$ .

Using the Peano–Baker series, if necessary, the solutions of the above matrix differential equations are given by $\Psi_t(s,s_0)=e^{A(t)(s-s_0)}$ and

\begin{align*}\Psi_{N,t}^0(s,s_0) = \textbf{1}_p+\int_{s_0}^s A\bigg(\frac{\tau_1}{N}+t\bigg) d\tau_1+ \int_{s_0}^s A\bigg(\frac{\tau_1}{N}+t\bigg) \int_{s_0}^{\tau_1} A\bigg(\frac{\tau_2}{N}+t\bigg) d\tau_2 d\tau_1 + \ldots.\end{align*}

Proof. Consider $Y_N(t)$ , $g_N^0$ , g, $\Psi_{N,t}^0$ , and $\Psi_t$ as defined above. For fixed $u,t\in\mathbb{R}$ it holds that

\begin{align*}\left| g_N^0(Nt,-u)-g(t,-u) \right|&= \mathbb{1}_{\{u\leq 0\}} \bigg| B(t)' \Big( \Psi_{N,t}^0(0,u) - \Psi_t(0,u) \Big) C\bigg(\frac{u}{N}+t\bigg) \\[5pt] &\qquad\qquad + B(t)' \Psi_t(0,u) \bigg( C\bigg(\frac{u}{N}+t\bigg) - C(t) \bigg)\bigg| \\[5pt] &\leq \mathbb{1}_{\{u\leq 0\}}\Big( \big\lVert B(t) \big\rVert \Big\lVert \Psi_{N,t}^0(0,u) - \Psi_t(0,u) \Big\rVert \bigg(\sup_{s\in\mathbb{R}}\big\lVert C(s) \big\rVert\bigg)\\[5pt] &\qquad\qquad+ \big\lVert B(t) \big\rVert\big\lVert \Psi_t(0,u) \big\rVert\bigg\lVert C\bigg(\frac{u}{N}+t) - C(t) \bigg\rVert\bigg) \;=\!:\;P_1+P_2.\end{align*}

Since $C(\!\cdot\!)$ is continuous, we immediately obtain $P_2\rightarrow0$ as $N\rightarrow\infty$ . In view of $P_1$ , it is sufficient to show that for all $\varepsilon>0$ and sufficiently large N,

(25) \begin{align}\Big\lVert \Psi_{N,t}^0(0,u)-\Psi_t(0,u) \Big\rVert\leq \varepsilon.\end{align}

Thanks to the equivalence of all norms on $M_{p\times p}(\mathbb{R})$ , it is sufficient to prove (25) for the norm of each column. By $\Psi_{N,t}^{0(j)}(0,u)$ and $\Psi_t^{(j)}(0,u)$ we denote the jth column, $j=1,\ldots,p$ , of $\Psi_{N,t}^0(0,u)$ and $\Psi_t(0,u)$ . Then, for functions $f_{N,t}(s,x)=A(\frac{s}{N}+t)x$ and $\tilde{f}_{t}(s,x)=A(t)x$ , we obtain

\begin{align*}\Psi_{N,t}^{0(j)}(u,u) &=e_j, \quad &&\frac{d}{ds} \Psi_{N,t}^{0(j)}(s,u) = f_{N,t}\Big(s,\Psi_{N,t}^{0(j)}(s,u)\Big),\\[5pt] \Psi_t^{(j)}(u,u)&= e_j, \quad \text{and} \quad && \frac{d}{ds} \Psi_t^{(j)}(s,u)= \tilde{f}_{t}\Big(s,\Psi_t^{(j)}(s,u)\Big), \end{align*}

where $e_j$ denotes the jth unit vector. Note that $f_{N,t}$ and $\tilde{f}_{t}$ are Lipschitz continuous in the second argument, with Lipschitz constant

\begin{align*} L=\sup_{s\in[u,0]}\bigg\lVert A\bigg(\frac{s}{N}+t\bigg) \bigg\rVert+A(t)<\infty.\end{align*}

Moreover,

\begin{align*}\Big\lVert f_t\Big(s,\Psi_{t}^{(j)}(s,u)\Big) - f_{N,t}\Big(s,\Psi_{t}^{(j)}(s,u)\Big) \Big\rVert \leq \delta \Big\lVert \Psi_{t}^{(j)}(s,u) \Big\rVert \leq \delta c,\quad s\in[u,0],\end{align*}

since $\left\lVert A(\frac{s}{N}+t)-A(t) \right\rVert<\delta$ for any $\delta>0$ for sufficiently large N, and $\Psi_{t}^{(j)}(\cdot,u)$ is continuous and thus bounded on [u,0]. An application of [Reference Walter46, Section 12.V] gives (25). Finally, by using the dominated convergence theorem with majorant

\begin{align*}\Big| g_N^0(Nt,-u) \Big| &\leq \mathbb{1}_{\{u\leq 0\}} \big\lVert B(t) \big\rVert \bigg(\sup_{s\in\mathbb{R}}\big\lVert C(s) \big\rVert\bigg) \Big\lVert \Psi_{N,t}^0(0,u) \Big\rVert \\[5pt] &\leq \mathbb{1}_{\{u\leq 0\}} c_t F_t(u)\in L^2(\mathbb{R})\end{align*}

for some constant $c_t>0$ , where the last inequality follows from (C1) and (C2), we can deduce that $\Big\lVert g_N^0(Nt,\cdot)-g(t,\cdot) \Big\rVert_{L^2}\rightarrow0$ as $N\rightarrow\infty$ .

In fact, the assumption (C3) in Proposition 9 is an immediate consequence if the state space system is uniformly exponentially stable.

Definition 11. ([Reference Rugh38, Chapter 6, Definition 6.5 and Theorem 6.7].) A sequence of linear state space models as in Definition 10 is called uniformly exponentially stable if there exist $\gamma>0$ and $\lambda>0$ such that

\begin{align*}\Big\lVert \Psi_{N,t}^0(s,s_0) \Big\rVert \leq \gamma e^{-\lambda(s-s_0)}\end{align*}

for all $s,s_0$ , where $s>s_0$ , $N\in\mathbb{N}$ , and $t\in\mathbb{R}$ .

If a linear state space model as in Definition 10 is uniformly exponentially stable, then the condition (C3) from Proposition 9 is satisfied for $F_t(u)\;:\!=\;\gamma e^{\lambda u}$ and some $\gamma,\lambda>0$ .

Remark 6. There is a strong structural resemblance between the above results and known results from the theory on locally stationary processes in discrete time. Indeed, let us consider a sequence of time-varying AR(p) processes given by $X_{t,T}=\sum_{j=1}^pa_j(t/T)X_{t-j,T}+\sigma(t/T)\varepsilon_t$ , where $\varepsilon_t$ is a sequence of i.i.d. random variables such that $\mathbb{E}[\varepsilon_t]=0$ and $Var(\varepsilon_t)=1$ , $a_j(u)$ is a family of arbitrary coefficient functions, and $\sigma(u)$ is an arbitrary function. For a comprehensive discussion on the sequence of processes $X_{t,T}$ we refer to [Reference Dahlhaus11]. In the following we consider the matrix function

\begin{align*}A(u)&\;:\!=\;\left( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}a_1(u) & a_2(u) & \ldots & \ldots & a_p(u) \\[5pt] 1 & 0 & \ldots & \ldots & 0 \\[5pt] 0 & 1 & \ddots & \ldots & \vdots \\[5pt] \vdots & \vdots & \ddots &\ddots & \vdots \\[5pt] 0 & \ldots & 0 & 1 & 0 \\[5pt] \end{array}\right).\end{align*}

In this setting, the authors of [Reference Dahlhaus11, Reference Künsch23] show that $X_{t,T}$ is locally stationary if $a_j(u)$ is continuous for all $j=1,\ldots,p$ and the eigenvalues $\lambda_i(u)$ , $i=1,\ldots,p$ , of A(u) satisfy $|\lambda_i(u)|\leq\delta$ for some $\delta\in(0,1)$ . Moreover, in [Reference Dahlhaus11, Theorem 2.3(ii)], the authors consider the $a_j(u)$ to be parameterized and derive suitable properties for the treatment of maximum likelihood estimates under the additional assumption that the $a_j(u)$ as well as their derivatives are uniformly bounded.

If the commutativity assumption (14) holds, the transition matrix is given by

\begin{align*} \Psi_{N,t}^0(0,u)=e^{\int_u^{0} A\left(\frac{s}{N}+t\right) ds}.\end{align*}

Then $Y_N(t)$ simplifies to

\begin{align*}Y_N(t) = \int_{-\infty}^{Nt} B(t)' e^{\int_{-(Nt-u)}^{0} A(\frac{s}{N}+t) ds} C\bigg(\frac{u}{N}\bigg) L(du).\end{align*}

Proof. It is sufficient to check (C3) from Proposition 9. We start by assuming that (D1) holds. Then, by [Reference Horn and Johnson19, Theorem 1.3.12], $\{A(t)\}_{t\in\mathbb{R}}$ is simultaneously diagonalizable. Thus, there exists a non-singular matrix S such that $S^{-1}A(\frac{s}{N}+t)S=diag(\lambda_1(\frac{s}{N}+t),\ldots,\lambda_p(\frac{s}{N}+t))\;=\!: D(\frac{s}{N}+t)$ . Considering the spectral norm, we obtain for all $u\leq0$

\begin{align*}\Big\lVert \Psi_{N,t}^0(0,u) \Big\rVert&=\Big\lVert e^{\int_u^{0} A(\frac{s}{N}+t) ds} \Big\rVert=\Big\lVert e^{\int_u^{0}S D(\frac{s}{N}+t) dsS^{-1}} \Big\rVert\leq\big\lVert S \big\rVert\Big\lVert S^{-1} \Big\rVert \Big\lVert e^{\int_u^{0}D(\frac{s}{N}+t) ds} \Big\rVert\\[5pt]&\leq C \max \bigg\{ \sqrt{\mu}\;:\; \mu\in\sigma\bigg( e^{\int_u^0 D(\frac{s}{N}+t)^* + D(\frac{s}{N}+t) ds} \bigg) \bigg\}\\[5pt] &= C \max_{j=1,\ldots,p} \sqrt{ e^{2\int_u^0 \mathfrak{Re}(\lambda_j(\frac{s}{N}+t)) ds} }\\[5pt] &= C \max_{j=1,\ldots,p} e^{\int_u^0 \mathfrak{Re}(\lambda_j(\frac{s}{N}+t)) ds} \leq C e^{\int_u^0 -\mu ds} = C e^{\mu u}\end{align*}

for some constant $C>0$ .

In the case where (D2) holds, we have

\begin{align*}\Big\lVert \Psi_{N,t}^0(0,u) \Big\rVert&=\Big\lVert e^{\int_u^{0} A(\frac{s}{N}+t) ds} \Big\rVert=\Big\lVert e^{\int_0^{-u} A(\frac{-s}{N}+t) ds} \Big\rVert\\[5pt] &\leq \mathbb{1}_{\left\{-u\in[0,\tau]\right\}}e^{\left| \tau \right|\sup_{s\in[0,\tau]}\left\lVert A\left(\frac{-s}{N}+t\right) \right\rVert}\\[5pt] &\quad+ \mathbb{1}_{\{-u> \tau\}}e^{\left| \tau \right|\sup_{s\in[0,\tau]}\left\lVert A\left(\frac{-s}{N}+t\right) \right\rVert} \Big\lVert e^{\int_\tau^{-u}A\left(\frac{-s}{N}+t\right)ds} \Big\rVert,\end{align*}

where we used that the integrals $\int_0^{\tau}A\left(\frac{-s}{N}+t\right)ds$ and $\int_\tau^{-u}A\left(\frac{-s}{N}+t\right)ds$ commute. Therefore, it is sufficient to bound

\begin{align*} \Big\lVert e^{\int_\tau^{-u}A(\frac{-s}{N}+t)ds} \Big\rVert.\end{align*}

In the following we use [Reference Lukes26, Theorem 7.7.1]. Since the family $\{A(t)\}_{t\in\mathbb{R}}$ is mutually commutative, the family can be reduced simultaneously to an upper triangular form by a single unitary transformation, i.e. there exists a unitary matrix $U\in M_{p\times p}(\mathbb{C})$ such that $U^*A(t)U=T(t)$ is an upper triangular matrix for all $t\in\mathbb{R}$ (see [Reference Horn and Johnson19, Theorem 2.3.3]). For each $x>\tau$ , the diagonal entries (and hence also the eigenvalues) of $\frac{1}{x}\int_\tau^xT\left( \frac{s}{N}+t \right)ds$ are $\frac{1}{x}\int_\tau^x \lambda_1\left(\frac{s}{N}+t\right)ds,\ldots,\frac{1}{x}\int_\tau^x \lambda_p\left(\frac{s}{N}+t\right)ds$ . Moreover, these are also the eigenvalues of $\frac{1}{x}\int_\tau^x A\left( \frac{s}{N}+t \right)ds$ , since

\begin{align*}\frac{1}{x}\int_\tau^xT\bigg( \frac{s}{N}+t \bigg)ds=\frac{1}{x}U^*\int_\tau^x A\bigg( \frac{s}{N}+t \bigg)ds\ U.\end{align*}

For the real part of the eigenvalues we obtain

\begin{align*}\mathfrak{Re}\bigg(\frac{1}{x}\int_\tau^x \lambda_i\bigg(\frac{s}{N}+t\bigg)ds\bigg)=\frac{1}{x}\int_\tau^x \mathfrak{Re}\bigg(\lambda_i\bigg(\frac{s}{N}+t\bigg)\bigg)ds\leq-\mu \bigg(\frac{x-\tau}{x}\bigg)\leq-\mu\end{align*}

for all $i=1,\ldots,p$ , $N\in\mathbb{N}$ , and $x\in\mathbb{R}$ . Hence

\begin{align*}\overline{\bigcup_{\tau\leq t<\infty}\!\!\!\tilde\sigma\bigg(\frac{1}{x}\int_\tau^x A\bigg( \frac{s}{N}+t \bigg)ds\bigg)}\subset\!\!\overline{\bigcup_{\tau\leq t<\infty}\!\!\!\sigma\bigg(\frac{1}{x}\int_\tau^x A\bigg( \frac{s}{N}+t \bigg)ds\bigg)}\!\subset \big\{z\in\mathbb{C}\;:\;\mathfrak{Re}(z)\leq-\mu\big\}\end{align*}

for all $N\in\mathbb{N}$ , where $\tilde\sigma(B)$ denotes the collection of all distinct eigenvalues of the matrix B and $\sigma(B)$ the spectrum of B. Finally, an application of [Reference Lukes26, Theorem 7.7.1] gives

\begin{align*}\Big\lVert e^{\int_\tau^{-u}A\left(\frac{-s}{N}+t\right)ds} \Big\rVert\leq Ce^{\mu u}\end{align*}

for some constant $C>0$ .

Sequences of time-varying CARMA processes, i.e. where the family $\{\mathcal{A}(t)\}_{t\in\mathbb{R}}$ forms a family of companion matrices, cannot be covered by Proposition 11, since companion matrices are in general not commutative. The following proposition brings further insight when a family of companion matrices is mutually commutative.

If, for a sequence of time-varying CARMA processes, the family $\{\mathcal{A}(t)\}_{t\in\mathbb{R}}$ is not mutually commutative, Proposition 9 provides sufficient conditions for local stationarity, where the condition (C3) can be derived from Proposition 10.