supCARMA Processes in Time Series Modeling

• supCARMA processes are superpositions of Lévy-driven CARMA processes defined via a Lévy basis, offering a flexible framework for modeling complex time series dynamics.
• They extend the supOU framework by incorporating state-space representations and kernel functions to capture long memory and oscillatory correlation structures.
• Applications include financial volatility, turbulence, and environmental data analysis, with simulation and estimation methods leveraging eigenstructure and spectral analysis.

SupCARMA processes are defined as superpositions of Lévy-driven Continuous-time AutoRegressive Moving Average (CARMA) processes with respect to a Lévy basis. They generalize the concept of superpositions of Ornstein-Uhlenbeck (supOU) type processes, resulting in a highly flexible class of stationary, infinitely divisible processes capable of modeling time series with complex second-order properties, such as long-range dependence and oscillatory correlation functions (Grahovac et al., 22 Jan 2026).

1. Formal Construction of supCARMA Processes

Let $A_p$ denote the set of $p \times p$ companion matrices,

$$A= \begin{pmatrix} 0 & 1 & 0 & \cdots & 0 \ 0 & 0 & 1 & \cdots & 0 \ \vdots & & & \ddots & \vdots \ 0 & 0 & 0 & \cdots & 1 \ -a_p & -a_{p-1} & \cdots & -a_1 \end{pmatrix}$$

with all eigenvalues in the open left half-plane to ensure stability. The vector $\bm e=(0,\dots,0,1)^\top$ and $\bm b=(b_0, \dots, b_{p-1})^\top$ ($b_j=0$ for $j>q$) are used to extract scalar output.

A Lévy basis $\Lambda$ on $A_p \times \mathbb{R}$, with generating quadruple $(\gamma, \Sigma, \mu, \pi)$ and probability kernel $\pi$ on $A_p$, is used as the driving noise.

The supCARMA$(p,q)$ process $X(t)$ is defined as the mixed moving average

$$X(t) = \int_{A_p} \int_{-\infty}^t \bm b^\top e^{A(t-s)} \bm e \;\Lambda(dA, ds) = \int_{A_p} \int_{-\infty}^t g(A, t-s)\;\Lambda(dA, ds),$$

where the impulse-response kernel is $$g(A,u) = \bm b^\top e^{A u} \bm e\, 1_{(0, \infty)}(u)$$.

In state-space terms, $$\bm X(t) = \int_{A_p} \int_{-\infty}^t e^{A(t-s)}\bm e\,\Lambda(dA, ds)$$ is a $p$-variate supOU process, with $X(t) = \bm b^\top \bm X(t)$.

2. Existence and Stationarity Conditions

Strict stationarity and infinite divisibility are guaranteed under transparent integrability requirements:

• Sufficient conditions from multivariate supOU theory:

$$\int_{|x|>1} \log|x|\,\mu(dx) < \infty, \quad \exists\,\rho(A) > 0,\,\eta(A) \ge 1:\,\|e^{As}\| \le \eta(A) e^{-\rho(A) s}, \quad \int_{A_p} \frac{\eta(A)^2}{\rho(A)}\,\pi(dA) < \infty.$$

• Rajput–Rosinski conditions for the kernel $f(A,u) = g(A,u)$:
  • $$\int_{A_p} \int_0^\infty |g(A,u)|\,du\,\pi(dA) < \infty$$
  • $$\int_{A_p} \int_0^\infty g(A,u)^2\,du\,\pi(dA) < \infty$$
  • $$\int_{A_p} \int_0^\infty \int_{\mathbb{R}} (1 \wedge |xg(A,u)|^2)\,\mu(dx)\,du\,\pi(dA) < \infty$$

Fulfillment ensures that $X$ is strictly stationary and infinitely divisible.

3. Kernel and Frequency-Domain Properties

The impulse-response kernel governing each CARMA contribution is

$$g(A,u) = \bm b^\top e^{A u}\bm e\,1_{(0,\infty)}(u).$$

The associated transfer function in the frequency domain is

$$H(A, i\omega) = \bm b^\top (i\omega I - A)^{-1} \bm e,$$

yielding the spectral density of $X$,

$$f_X(\omega) = \frac{1}{2\pi} \int_{A_p} |H(A, i\omega)|^2 \left( \Sigma + \int x^2\,\mu(dx) \right)\,\pi(dA).$$

4. Classification and Properties of supCAR(2) Processes

SupCAR$(2)$ processes admit a detailed classification via the eigenstructure of the underlying $2 \times 2$ CAR(2) matrix: $$A = \begin{pmatrix} 0 & 1 \ -a_2 & -a_1 \end{pmatrix}, \quad \xi_{1,2} = \frac{-a_1 \pm \sqrt{a_1^2 - 4a_2}}{2}$$ Three canonical cases:

Type	Eigenstructure	Impulse-Response Kernel $g(u)$
I	Double real root $\xi_1 = \xi_2 = -a/2$	$g_I(a, u) = u\,e^{-a u/2} 1_{u > 0}$
II	Distinct real roots $\xi_1 = -\lambda,\, \xi_2 = -\lambda \theta$	$$g_{II}(\lambda, \theta, u) = \frac{e^{-\lambda \theta u} - e^{-\lambda u}}{\lambda (1-\theta)} 1_{u > 0}$$
III	Complex conjugates $r e^{\pm i \psi}$, $r > 0$, $\psi \in (\pi/2, \pi)$	$$g_{III}(r, \psi, u) = \frac{e^{r u \cos\psi}}{r\sin\psi} \sin(r u \sin\psi)\,1_{u>0}$$

This partitioning yields markedly distinct second-order and spectral properties, including non-monotone and oscillatory autocorrelation functions.

5. Correlation Structure and Long-Range Dependence

Under finite second moment conditions, supCAR(2) subclass correlations are:

• Type I: $$r_I(\tau) = \frac{ \int_0^\infty a^{-3} (\frac{a}{2} \tau + 1) e^{-a \tau/2} \pi_I(da) }{ \int_0^\infty a^{-3} \pi_I(da) }$$ When $\pi_I(da) \propto a^{\alpha+2}e^{-a}da$, $r_I(\tau) \sim C\tau^{-\alpha}$ exhibits power-law decay for $\alpha\in(0,1]$.
• Type II: $$r_{II}(\tau) = \frac{ \int \lambda^{-3} (1-\theta^2)^{-1} \left( \theta^{-1}e^{-\lambda\theta\tau} - e^{-\lambda\tau} \right) \pi_{\lambda,\theta}(d\lambda, d\theta)}{2 \int \lambda^{-3}\theta^{-1}(1+\theta)^{-1}\,\pi_{\lambda,\theta}(d\lambda,d\theta)}$$
• Type III: $$r_{III}(\tau) = \frac{ (L(1)) \int r^{-3} \frac{e^{r\tau\cos\psi}}{2\sin 2\psi} \sin(r\tau\sin\psi-\psi)\,\pi_{r,\psi}(dr, d\psi)}{2 (X_{III}(0))}$$ A plausible implication is that regularly varying $\pi_r$ densities at zero ensure power-law decay, while the sine term induces oscillatory behavior in the correlation function.

6. Spectral Density and Oscillatory Dynamics

The spectral density formulation for mixed–moving–average processes is

$$f_X(\omega) = \frac{1}{2\pi} \int |\bm b^\top (i\omega I-A)^{-1}\bm e|^2 \left( \Sigma + \int x^2\mu(dx)\right) \pi(dA).$$

In supCAR(2)-Type III, complex eigenvalues $r e^{\pm i\psi}$ generate damped oscillations in both autocovariance and spectral density, supporting time series exhibiting quasi-periodic features and intricate frequency structures.

7. Computational Methods and Applied Contexts

Simulation proceeds by drawing i.i.d. matrices $A_i \sim \pi$, then simulating each CARMA$(p,q)$ component (using methods like Euler–Maruyama or exact filters), with aggregation via the Lévy basis.

Estimation approaches include empirical cumulant matching, generalized method of moments on autocovariances, and adaptations of multivariate supOU estimation frameworks.

Applications span modeling of financial volatility, turbulence, and environmental time series:

• Non-monotone or oscillatory correlation structures (Type III)
• Long-range dependence with tunable strength (Types I, II, III)
• Decoupling of marginal and dependence behavior via the Lévy basis and kernel $\pi$

This suggests supCARMA processes provide a rigorous and transparent extension of supOU methodology to accommodate richer second-order dynamics, oscillatory phenomena, and long memory effects in stationary stochastic modeling (Grahovac et al., 22 Jan 2026).