Noise recovery for L\'evy-driven CARMA processes and high-frequency behaviour of approximating Riemann sums

We consider high-frequency sampled continuous-time autoregressive moving average (CARMA) models driven by finite-variance zero-mean L\'evy processes. An L^2-consistent estimator for the increments of the driving L\'evy process without order selection in advance is proposed if the CARMA model is invertible. In the second part we analyse the high-frequency behaviour of approximating Riemann sum processes, which represent a natural way to simulate continuous-time moving average processes on a discrete grid. We shall compare their autocovariance structure with the one of sampled CARMA processes, where the rule of integration plays a crucial role. Moreover, new insight into the kernel estimation procedure of Brockwell et al. (2012a) is given.


Introduction
The constantly increasing availability of high-frequency data in finance and sciences in general has sparked in the last decade a great deal of attention about the asymptotic behaviour of highfrequency sampled processes, especially concerning the estimation of multi-power variations of Itō semimartingales (see, e.g., Andersen and Todorov (2010), Barndorff-Nielsen and Shephard (2003)), employing their realised counterparts. These quantities are of primary importance to practitioners, since they embody the deviation of data from a Brownian motion. Such methods are summarised in the book of Jacod and Protter (2012), which represents the most recent review on the subject.
In many areas of application Lévy-driven processes are used for modelling time series. An ample class within this group are continuous-time moving average (CMA) processes where g is the so-called kernel function and L = {L t } t∈R is said to be the driving Lévy process (see, e.g., Sato (1999) for a detailed introduction). They cover, for instance, Ornstein-Uhlenbeck and continuous-time autoregressive moving average (CARMA) processes. The latter are the continuous-time analogue of the well-known ARMA models (see, e.g., Brockwell and Davis (1991)) and have extensively been studied over the recent years (cf. Brockwell (2001Brockwell ( , 2004; Brockwell and Lindner (2009) ;Todorov and Tauchen (2006)). Originally, driving processes of CARMA models were restricted to Brownian motion (see Doob (1944), and also Doob (1990)). However, Brockwell (2001) allowed for Lévy processes with a finite rth moment for some r > 0.
In this paper we are concerned with a high-frequency sampled CARMA process driven by a second-order zero-mean Lévy process. Under the assumption of invertibility of the CARMA model, we present an L 2 -consistent estimator for the increments of the driving Lévy process, employing standard time series techniques. It is remarkable that the proposed procedure works for arbitrary autoregressive and moving average orders, i.e. there is no need for order selection in advance. In the light of the results in Brockwell et al. (2012a) and the flexibility of CARMA processes, the method might apply to a wider class of CMA models, too. Moreover, since the proof employes only the fact that the increments of the Lévy process are orthogonal rather than independent, the result holds for a much broader class of driving processes. Notable examples are the COGARCH processes (Brockwell et al. (2006); Klüppelberg et al. (2004)) or time-changed Lévy processes (Carr et al. (2003)), which are often used to model volatility clustering in finance and intermittency in turbulence.
This noise recovery result gives rise to the conjecture that the sampled CARMA process behaves on a high-frequency time grid approximately like a suitable MA(∞) model that we call approximating Riemann sum process. By comparing the asymptotic properties of the autocovariance structure of high-frequency sampled CARMA models with the one of their approximation Riemann sum processes, it will turn out that the so-called rule of the Riemann sums plays a crucial role if the difference between the autoregressive and moving average order is greater than one. On the one hand, this gives new insight into the kernel estimation procedure studied in Brockwell et al. (2012a) and explains at which points the kernel is indeed estimated. On the other hand, this has obvious consequences for simulation purposes. Riemann sum approximations are an easy tool to simulate CMA processes. However, our results show that one has to be careful with the chosen rule of integration in the context of certain CARMA processes.
The outline of the paper is as follows. In Section 2 we recall the definition of finite-variance CARMA models and summarise important properties of high-frequency sampled CARMA processes. In particular, we fix a global assumption that guarantees causality and invertibility for the sampled sequence. In the third section we then derive an L 2 -consistent estimator for the increments of the driving Lévy process starting from the Wold representation of the sampled process. It will turn out that invertibility of the original continuous-time model is sufficient and necessary for the recovery result to hold. Section 3 is completed by an illustrating example for CAR(2) and CARMA(2, 1) processes. Thereafter, the high-frequency behaviour of approximating Riemann sum processes is studied in Section 4. First, an ARMA representation for the Riemann sum approximation is established in general and then the role of the rule of integration is analysed by matching the asymptotic autocovariance structure of sampled CARMA processes and their Riemann sum approximations in the cases where the autoregressive order is less or equal to three. The connection between the Wold representation and the approximating Riemann sum yields a deeper insight into the kernel estimation procedure introduced in Brockwell et al. (2012a). The proof of Theorem 3.2 and some auxiliary results can be found in the appendix.

Finite-variance CARMA processes
Throughout this paper we are concerned with a CARMA process driven by a second-order zeromean Lévy process L = {L t } t∈R with EL 1 = 0 and EL 2 1 = 1. It is defined as follows. For non-negative integers p and q such that q < p, a CARMA(p, q) process Y = {Y t } t∈R with real coefficients a 1 , . . . , a p , b 0 , . . . , b q and driving Lévy process L is defined to be a strictly stationary solution of the suitably interpreted formal equation where D denotes differentiation with respect to t, a(·) and b(·) are the characteristic polynomials, a(z) := z p + a 1 z p−1 + · · · + a p and b(z) := b 0 + b 1 z + · · · + b p−1 z p−1 , the coefficients b j satisfy b q = 1 and b j = 0 for q < j < p, and σ is a positive constant. The polynomials a(·) and b(·) are assumed to have no common zeroes. We denote, respectively, by λ i and −µ i the roots of a(·) and b(·), such that these polynomials can be written as a(z) = p i=1 (z − λ i ) and b(z) = q i=1 (z + µ i ). Moreover, we suppose permanently Assumption 1. (i) The zeroes of the polynomial a(·) satisfy (λ j ) < 0 for every j = 1, . . . , p, (ii) and the roots of b(·) have non-vanishing real part, i.e. (µ j ) = 0 for all j = 1, . . . , q.
Since the derivative DL t does not exist in the usual sense, we interpret (2.1) as being equivalent to the observation and state equations It is easy to check that the eigenvalues of the matrix A are the same as the zeroes of the autoregressive polynomial a(·).
Under Assumption 1(i) it has been shown in (Brockwell and Lindner (2009), Theorem 3.3) that Eqs. (2.2)-(2.3) have the unique strictly stationary solution and ρ is any simple closed curve in the open left half of the complex plane encircling the zeroes of a(·). The sum is over the distinct zeroes λ of a(·) and Res z=λ (·) denotes the residue at λ of the function in brackets. The kernel g can be expressed (cf. Brockwell and Lindner (2009), Equations (2.10) and (3.7)) also as g(t) = σb e At e p 1 (0,∞) (t), t ∈ R, (2.6) and its Fourier transform is , ω ∈ R. (2.7) In the light of Eqs. (2.4)-(2.7), we can interpret a CARMA process as a continuous-time filtered white noise whose transfer function has a finite number of poles and zeroes. We emphasise that the condition on the roots of a(·) to lie in the interior of the left half of the complex plane in order to have causality arises from Theorem V, p. 8, Paley and Wiener (1934), which is intrinsically connected with the theorems in Titchmarsh (1948), pp. 125-129, on the Hilbert transform. A similar request on the roots of b(·) will turn out to be necessary for recovering the driving Lévy process.

Properties of high-frequency sampled CARMA processes
We now recall some properties of the sampled sequence Y ∆ := {Y n∆ } n∈Z of a CARMA(p, q) process where ∆ > 0; cf. Brockwell et al. (2012a,b) and references therein. It is known that the sampled process Y ∆ satisfies the ARMA(p, p − 1) equations where B is the discrete-time backshift operator, BY ∆ n := Y ∆ n−1 . Finally, the MA part Θ ∆ (·) is a polynomial of order p − 1, chosen in such a way that it has no roots inside the unit circle. For p > 3 and fixed ∆ > 0 there is no explicit expression for the coefficients of Θ ∆ (·) nor the white noise process Z ∆ . Nonetheless, asymptotic expressions for Θ ∆ (·) and σ 2 ∆ = var(Z ∆ n ) as ∆ ↓ 0 were obtained in Brockwell et al. (2012a,b). Namely, we have that the polynomial Θ ∆ (z) and the variance σ 2 ∆ can be written as (see Theorem 2.1, Brockwell et al. (2012a)) (2.9) where, again as ∆ ↓ 0, The signs ± in (2.11) are chosen in such a way that, for sufficiently small ∆, the coefficients ζ k and η(ξ i ) are less than one in absolute value. This ensures that Eq. (2.8) is invertible. Moreover, ξ i are the zeroes of the function α p−q−1 (·) that is defined as the (p − q − 1)-th coefficient in the series expansion where the LHS of Eq. (2.12) is a power transfer function arising from the sampling procedure (cf. Brockwell et al. (2012b), Eq. (11)). Therefore the coefficients η(ξ i ) can be regarded as spurious since they do not depend on the parameters of the underlying continuous-time process Y , but just on p − q.
Remark 2.1. Our notion of sampled process is a weak one since we require only that the sampled sequence has the same autocovariance structure as the continuous-time model observed on a discrete grid. We know that the filtered process on the LHS of (2.8) (Brockwell and Lindner (2009), Lemma 2.1) is a (p − 1)-dependent discrete-time process. Therefore there exist 2 p−1 possible representations for the RHS of (2.8), each yielding the same autocovariance function of the filtered process, but only one has its roots outside the unit circle. The latter is called minimumphase spectral factor (see Sayed and Kailath (2001) for a review on the topic). Since it is not possible to discriminate between the different factorisations, we always take the minimum-phase spectral factor without any further question. This will be crucial for our main result. Moreover, the rationale behind Assumption 1(ii) becomes clear now: if (µ k ) = 0 for some k, then the corresponding |ζ k | 2 is equal to 1 + ∆ 2 |µ k | 2 + o(∆ 2 ) for either sign choice. In this case, the MA(p − 1) polynomial in Eq. (2.9) cannot be invertible for small ∆.
To ensure that the sampled CARMA process is invertible, we need to verify that |η(ξ i )| is strictly less than one for sufficiently small ∆.

Noise recovery
In this section we prove the first main statement of the paper, a recovery result for the driving Lévy process. We start with some motivation for our approach.
We know that the sampled CARMA sequence Y ∆ = {Y n∆ } n∈Z has the Wold representation (cf. Brockwell and Davis (1991) is the causal representation of Eq. (2.8), and it has been shown in Brockwell et al. (2012a) that for every causal and invertible CARMA(p, q) process, as where g is the kernel in the moving average representation (2.4). Given the availability of classical time series methods to estimate {ψ ∆ j } j∈N and σ 2 ∆ , and the flexibility of CARMA processes, we argue that this result can be applied to more general continuous-time moving average models.
In view of Eqs. (3.1) and (3.2) it is natural to investigate whether the quantitȳ approximates the increments of the driving Lévy process in the sense that for every fixed t > 0, 3) The first results on retrieving the increments of L were given in Brockwell et al. (2011), and further generalized to the multivariate case by Brockwell and Schlemm (2013). The essential limitation of this parametric method is that it might not be robust with respect to model misspecification. More precisely, the fact that a CARMA(p, q) process is (p − q − 1)-times differentiable (see Proposition 3.32 of Marquardt and Stelzer (2007)) is crucial for the procedure to work (cf. Theorem 4.3 of Brockwell and Schlemm (2013)). However, if the underlying process is instead CARMA(p , q ) with p − q < p − q, then some of the necessary derivatives do not exist anymore. In contrast to the aforementioned procedure, in the method we propose there is no need to specify the autoregressive and the moving average orders p and q in advance.
Before we start to prove the recovery result in Eq. (3.3), let us establish the notion of invertibility in analogy to the discrete-time case.
Definition 3.1. A CARMA(p, q) process is said to be invertible if the roots of the moving average polynomial b(·) have negative real parts, i.e. (µ i ) > 0 for all i = 1, . . . , q.
Our main theorem is the following. Its proof can be found in the appendix.
Theorem 3.2. Let Y be a finite-variance CARMA(p, q) process and Z ∆ the noise on the RHS of the sampled Eq. (2.8). Moreover, let Assumption 1 hold and defineL ∆ : if and only if the roots of the moving average polynomial b(·) on the RHS of the CARMA Eq. (2.1) have negative real parts, i.e. if and only if the CARMA process is invertible.
Remark 3.3. (i) It is an easy consequence of the triangle and Hölder's inequality that, if the recovery result (3.4) holds, then also (ii) Minor modifications of the proof of Theorem 3.2 show that the recovery result in Eq. (3.4) remains still valid if we drop the assumption of causality, Assumption 1(i), and suppose instead only (λ j ) = 0 for every j. Hence, invertibility of the CARMA process is necessary for the noise recovery result to hold, whereas causality is not. Note that the white noise process in the non-causal case is not the same as in the Wold representation (3.1).
(iii) The necessity and sufficiency of the invertibility assumption descends directly from the fact that we choose always the minimum-phase spectral factor as pointed out in Remark 2.1.
(iv) The proof of Theorem 3.2 suggests that this procedure should work in a much more general framework. Let I ∆ (·) denote the inversion filter in Eq. (A.1) and ψ ∆ := ψ ∆ i i∈N the coefficients in the Wold representation (3.1). The proof essentially needs, apart from the rather technical Lemma A.3, that, as ∆ ↓ 0, provided that the function ∞ k=0 ψ ∆ k z k does not have any zero inside the unit circle. In other words, we need that the discrete Fourier transform in the denominator of Eq. (3.5) converges to the Fourier transform in the numerator; this can be easily related to the kernel estimation result in Eq. (3.2). Given the peculiar structure of CARMA processes, this relationship can be calculated explicitly, but the results should hold true for continuous-time moving average models with more general kernels, too.
We illustrate Theorem 3.2 and the necessity of the invertibility assumption by an example where the convergence result is established using a time domain approach. That gives an explicit result also when the invertibility assumption is violated.
Unfortunately this strategy is not viable for a general CARMA process due to the complexity of involved calculations when p is greater than two.
Example 3.4 (CARMA(2, q) process). The causal CARMA(2, q) process is the strictly stationary solution to the formal stochastic differential equation where λ 1 , λ 2 < 0, λ 1 = λ 2 and b ∈ R\{0}. It can be represented as a continuous-time moving average process as in Eq. (2.4), with kernel function for t > 0 and 0 elsewhere. The corresponding sampled process Y ∆ n = Y n∆ , n ∈ Z, satisfies the causal and invertible ARMA(2, 1) equations as in (2.8). From Eq. (27) of Brockwell et al. (2012b) we know for any n ∈ Z that Inversion of Eq. (2.8) gives, for every ∆ > 0, The sequence Z ∆ := {Z ∆ n } n∈Z is a weak white noise process. Moreover, using ∆L n = n∆ (n−1)∆ dL s , we observe that For any fixed t ∈ (0, ∞), since ∆L andL ∆ are both second-order stationary white noises with variance ∆, we obtain that where the last equality is deduced from Eq. (3.6). For every a = 1, and the variance of the error can be explicitly calculated as We now compute the asymptotic expansion for ∆ ↓ 0 of the equation above. We obviously have that 2 t/∆ ∆ = 2t(1 + o(1)) and, using the explicit formulas for the kernel functions g, Hence, for a fixed t ∈ (0, ∞) and ∆ ↓ 0, we get i.e. (3.4) holds always for q = 0, whereas for q = 1 if and only if b > 0. If b < 0, the error made by approximating the driving Lévy by inversion of the discretised process grows as 4t for large t.

High-frequency behaviour of approximating Riemann sums
The fact that, in the sense of Eq. (3.3),L ∆ n ≈ ∆L n = L n∆ − L (n−1)∆ for small ∆, along with Eq. (3.2), gives rise to the conjecture that the Wold representation for Y ∆ behaves on a highfrequency time grid approximately like the MA(∞) process with some h ∈ [0, 1] and g is the kernel function as in (2.6). In other terms, we have for a CARMA process, under the assumption of invertibility and causality, that the discrete-time quantities appearing in the Wold representation approximate the quantities in Eq. (4.1) when ∆ ↓ 0. The transfer function of Eq. (4.1) is defined as and its spectral density can be written as It is well known that a CMA process can be defined (for a fixed time point t) as the L 2 -limit of Eq. (4.1); this fact is naturally employed to simulate a CMA model when all the relevant quantities are known a priori. Therefore, we callỸ ∆,h approximating Riemann sum of Eq. (2.4), and h is said to be the rule of the approximating sum. If, for instance, h is chosen to be 1/2, we have the popular mid-point rule.
Remark 4.1. (i) It would be possible to consider more sophisticated integration rules by taking more nodes on every interval of length ∆ and suitable weights. However, since mostly used in practice, we decided to concentrate on that "simple" Riemann sum approximation.
(ii) In practice, when considering simulation studies for instance, one has to use a finite (truncated) Riemann sum of the form where N ∈ N is usually taken as a large number. If we let N = N (∆) → ∞ as ∆ → 0 with a suitable rate (N (∆) should diverge faster than ∆ goes to 0, e.g. N (∆) = ∆ −(1+ε) ), the main result of this section, Corollary 4.6, remains valid.
To give an answer to our conjecture, we investigate properties of the approximating Riemann sumỸ ∆,h of a CARMA process and compare its asymptotic autocovariance structure with the one of the sampled CARMA sequence Y ∆ . This yields more insight into the role of h for the behaviour ofỸ ∆,h as a process.
We start with a well-known property of approximating sums.
Proposition 4.2. Let g be in L 2 and Riemann-integrable. Then, for every h ∈ [0, 1], as ∆ ↓ 0: Proof. This follows immediately from the hypotheses made on g and the definition of L 2 -integrals.
This result essentially says only that approximating sums converge to Y t for every fixed time point t. However, for a CARMA(p, q) process we have that the approximating Riemann sum process satisfies for every h and ∆ an ARMA(p, p − 1) equation (see Proposition 4.3 below). This means that there might exist a process whose autocorrelation structure is the same as the one of the approximating sum. Given that the AR filter in this representation is the same as in Eq. (2.8), it is reasonable to investigate whether Φ ∆ (B)Y ∆ and Φ ∆ (B)Ỹ ∆,h have, as ∆ ↓ 0, the same asymptotic autocovariance structure, which can be expected but is not granted by Proposition 4.2.
The following proposition states the ARMA(p, p − 1) representation for the approximating Riemann sum. Proposition 4.3. Let Y be a CARMA(p, q) process, satisfying Assumption 1. Furthermore, suppose that the roots of the autoregressive polynomial a(·) are distinct. The approximating Riemann sum processỸ ∆,h of Y defined by Eq. (4.1) satisfies, for every h ∈ [0, 1], the ARMA(p, p − 1) equation The right-hand sum is defined to be one for k = 0 and it is evaluated over all possible subsets {j 1 , . . . , j k } of {1, . . . , p}\{l} with cardinality k, if k > 0.
Remark 4.4. (i) The approximating Riemann sum of a causal CARMA process is automatically a causal ARMA process. On the other hand, even if the CARMA model is invertible in the sense of Definition 3.1, the roots ofΘ ∆,h (·) may lie inside the unit circle, causing Y ∆,h to be non-invertible.
(ii) It is easy to see thatθ ∆,h 0 = g(h∆). If p − q ≥ 2 and h = 0, we have thatθ ∆,0 0 = 0, giving thatΘ ∆,0 (0) = 0. This is never the case for Θ ∆ (·) as one can see from Eq. (2.9) and Proposition 2.2. Moreover, it is possible to show that for h = 1 and p − q ≥ 2, the coefficientθ ∆,1 p−1 is equal to 0, implying that (4.3) is actually an ARMA(p, p − 2) equation. For those values of h, the ARMA equations solved by the approximating Riemann sums can never have the same asymptotic form as Eq. (2.8). Therefore, we restrict ourselves to the case h ∈ (0, 1) from now on.
(iii) The assumption of distinct autoregressive roots might seem restrictive, but the omitted cases can be obtained by letting distinct roots tend to each other. This would, of course, change the coefficients of the MA polynomial in Eq. (4.4). Moreover, as shown in Brockwell et al. (2012a,b), the multiplicity of the zeroes does not matter when L 2 -asymptotic relationships as ∆ ↓ 0 are considered.
Due to the complexity of retrieving the roots of a polynomial of arbitrary order from its coefficients, finding the asymptotic expression ofΘ ∆,h (·) for arbitrary p is a daunting task. Nonetheless, by using Proposition 4.3, it is not difficult to give an answer for processes with p ≤ 3, which are the most used in practice.
Proposition 4.5. LetỸ ∆,h be the approximating Riemann sum for a CARMA(p, q) process, suppose p ≤ 3, and let Assumption 1 hold and the roots of a(·) be distinct.
In general, the autocorrelation structure depends on h through the parameters χ p−q,i (h). In a time series context, it is reasonable to require that the approximating Riemann sum has the same asymptotic autocorrelation structure as the CARMA process that we want to approximate.
Corollary 4.6. Let the assumptions of Proposition 4.5 hold. Then Φ ∆ (B)Y ∆ and Φ ∆ (B)Ỹ ∆,h have the same asymptotic autocovariance structure as ∆ ↓ 0 Moreover, the MA polynomials in Eqs. (2.9) and (4.5) coincide if and only if the CARMA process is invertible and |χ p−q,i (h)| < 1, that is For p − q = 3, such an h does not exist.
To prove the second part of the corollary, we start observing that, under the assumption of an invertible CARMA process, the coefficients depending on µ i , if any, coincide automatically. Then it remains to check whether the coefficients depending on h can be smaller than 1 in absolute value. The cases p − q = 1, 2 follow immediately. Moreover, to see that there is no such h for p − q = 3, it is enough to notice that, for any h ∈ (0, 1), we have |χ 3,1 (h)| > 1 and 0 < |χ 3,2 (h)| < 1. Hence, they never satisfy the sought requirement for h ∈ (0, 1).
Remark 4.7. It is also feasible to use spectral densities rather than covariances in the proof of Corollary 4.6. In that case, one has to compare the spectral densities of Φ ∆ (B)Y ∆ and Φ ∆ (B)Ỹ ∆,h asymptotically as ∆ → 0. This would lead to the question whether the equation holds for any z ∈ C with |z| = 1 as ∆ → 0. Of course, (4.6) implies the same values for h as those stated in Corollary 4.6.
Corollary 4.6 can be interpreted as a criterion to choose an h such that the Riemann sum approximates the continuous-time process Y in a stronger sense than the simple convergence as a random variable for every fixed time point t. The second part of the corollary says that there is an even more restrictive way to choose h if we want Eqs. (2.9) and (4.5) to coincide. If the two processes satisfy asymptotically the same causal and invertible ARMA equation, they have the same coefficients in their Wold representations as ∆ ↓ 0. In the case of the approximating Riemann sum these coefficients are given explicitly by definition in Eq. (4.1).
In the light of Eq. (3.2) and Theorem 3.2, the sampled CARMA process behaves asymptotically like its approximating Riemann sum process for some specific h =h, which might not even exist as in the case p = 3, q = 0. However, if such anh exists, the kernel estimators (3.2) can be improved to For invertible CARMA(p, q) processes with p − q = 1, any choice of h would accomplish that. In principle anh can be found by matching a higher-order expansion in ∆, where higher-order terms depend on h.
For p−q = 2, there is only a specific value h =h := (3+ √ 3)/6 such thatỸ ∆,h behaves as Y ∆ in this particular sense. Therefore, it advocates for a unique, optimal value for, e.g., simulation purposes.
Finally, for p − q = 3, a similar value does not exist, meaning that it is not possible to mimic Y ∆ in this sense with any approximating Riemann sum.
To confirm these observations, we now give a small numerical study. We consider three different causal and invertible processes, a CARMA(2, 1), a CAR(2), and a CAR(3) model with parameters λ 1 = −0.7, λ 2 = −1.2, λ 3 = −2.6 and µ 1 = 3. Of course, for the CARMA(2, 1) we use only λ 1 , λ 2 and µ 1 , whereas for the CAR processes there is no need for µ 1 . We estimate the kernel functions from the theoretical autocorrelation functions using (3.2) as in Brockwell et al. (2012a). Our sampling rates are moderately high, namely 2 2 = 4 ( Figure 1) and 2 6 = 64 samplings per unit of time ( Figure 2). To see where the kernel is being estimated, we plot the kernel estimations on different grids. The small circles denote the extremal cases h = 0 and h = 1, the vertical sign the mid-point rule h = 0.5, and the diamond and the square are the values given in Corollary 4.6, if any. The true kernel function is then plotted with a solid, continuous line. For the sake of clarity, only the first eight estimates are plotted.
For the CARMA(2, 1) process, the kernel estimation seems to follow a mid-point rule (i.e. h = 1/2). For the CAR(2) process, the predicted valueh = (3 + √ 3)/6 (denoted with squares) is definitely the correct one, and for the CAR(3) the estimation is close for every h ∈ [0, 1], but constantly biased. In the limit ∆ ↓ 0, the slightly weaker results given by Eq. (3.2) still hold, showing that the bias vanishes in the limit. The conclusion expressed above is true for both considered sampling rates, which is remarkable since they are only moderately high in comparison with the chosen parameters.

A Proof of Theorem 3.2 and auxiliary results
Throughout the appendix, we use the same notation as in the preceding sections. We start with the proof of our main theorem in Section 3.  Proof of Theorem 3.2. Due to Assumption 1(ii) and Proposition 2.2, the sampled ARMA equation (2.8) is invertible. The noise on the RHS of Eq. (2.8) is then obtained using the classical inversion formula where B is the usual backshift operator. Let us consider the stationary continuous-time process The coefficients a ∆ i on the RHS of Eq. (A.1) are determined by the Laurent series expansion of the rational function Φ ∆ (·)Θ −1 ∆ (·). Moreover, Z ∆ n∆ = Z ∆ n for every n ∈ N; as a consequence, the random variables Z ∆ s , Z ∆ t are uncorrelated for |t − s| ≥ ∆ and var(Z ∆ t ) = var(Z ∆ n ). Exchanging the sum and the integral signs in Eq. (A.1), and since g(·) = 0 for negative arguments, we have that Z ∆ is a continuous-time moving average process whose kernel function g ∆ has Fourier transform (cf. Eq. (2.7)) Since we can write L t − L t−∆ = t −∞ 1 (0,∆) (t − s)dL s , the sum of the differences between the rescaled sampled noise terms and the increments of the Lévy process is given by where, for every n ∈ N, Note that the stochastic integral in Eq. (A.2) w.r.t. L is still in the L 2 -sense. It is a standard result, cf. (Gikhman and Skorokhod, 2004, Ch. IV, §4), that the variance of the moving average process in Eq. (A.2) is given by where the latter equality is true since h ∆ n (s) = 0 for any s ≤ 0. Furthermore, the Fourier transform of h ∆ n (·) can be readily calculated, invoking the linearity and the shift property of the Fourier transform. We thus obtain Due to Plancherel's Theorem, we deduce It is easy to see that the first two integrals in Eq. (A.3) are, respectively, the variances of n i=1L ∆ j and L n∆ , both equal to n∆. Setting n := t/∆ yields for fixed positive t, as ∆ ↓ 0, Hence, to show Eq. (3.4), it remains to prove that which in turn is equivalent to Now, Lemma A.2 asserts that the integrand in Eq. (A.4) converges pointwise, for every ω = 0, to 2(1 − cos(ωt))/ω 2 as ∆ ↓ 0. Since, for sufficiently small ∆, the integrand is dominated by an integrable function (see Lemma A.3), we can apply Lebesgue's Dominated Convergence Theorem and deduce that the LHS of Eq. (A.4) converges, as ∆ ↓ 0, to This proves (A.4) and concludes the proof of the "if"-statement.
In the following, we state three auxiliary results. For the proof of the first one, we need a concrete representation of the function α n (x), which is defined in Eq. (2.12). It can be shown that where P n (x) is a polynomial of order n in x, namely P n (x) = n j=0 x n−j n k=j+1 (2k)! 2n + 1 2k x n−j n k=j (2k + 1)! 2n + 1 2k + 1 with · · being the Stirling number of the second kind.
Proposition A.1. All the zeroes of α n (x) are real, distinct and greater than 2.
This shows Eq. (A.14) and thus concludes the proof.