On the sample autocovariance of a Lévy driven moving average process when sampled at a renewal sequence

Brandes, Dirk-Philip; Curato, Imma Valentina

doi:10.48550/arxiv.1804.02254

Cited by 3 publications

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“…In many applications different sampling schemes are also highly relevant and for some special cases results have been obtained. For example, [14] considers the asymptotics of the autocovariance function for Lévy-driven moving average processes sampled at an independent renewal sequence and [26] consider the asymptotics of the pathwise Fourier transform/periodogram for Lévy-driven CARMA processes sampled at deterministic irregular grids. Considering independent renewal sampled Lévy-driven MMA processes is beyond the scope of the present paper and the content of future research just starting in [15] where the preservation of strong mixing and weak dependence properties is discussed in general.…”

Section: R)-measurable Function With Values Inmentioning

confidence: 99%

Weak dependence and GMM estimation of supOU and mixed moving average processes

Curato¹,

Stelzer²

2019

Electron. J. Statist.

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We consider a mixed moving average (MMA) process X driven by a Lévy basis and prove that it is weakly dependent with rates computable in terms of the moving average kernel and the characteristic quadruple of the Lévy basis. Using this property, we show conditions ensuring that sample mean and autocovariances of X have a limiting normal distribution. We extend these results to stochastic volatility models and then investigate a Generalized Method of Moments estimator for the supOU process and the supOU stochastic volatility model after choosing a suitable distribution for the mean reversion parameter. For these estimators, we analyze the asymptotic behavior in detail.An important example of MMA are the supOU processes studied in [3,10,25,27]. In the univariate case, assume |x|>1 log(|x|) ν(dx) < ∞ and R − − 1 A π(dA) < ∞, where ν is a Lévy measure and π is the probability distribution on R − of the random parameter A, see Definition 2.1 for details. If Λ is a Lévy basis on R with those characteristics, then the processis called a supOU process. Whereas a non-Gaussian Ornstein-Uhlenbeck process necessarily exhibits autocorrelation e ah for h ∈ N, the supOU process has a flexible dependence structure. For example, its autocorrelations can show a polynomial decay depending on the probability distribution π. Moreover, when a discrete probability distribution π for the random parameter A is considered, we obtain a popular model used, for example, in stochastic volatility models [3], in modeling fractal activity times [36,37] and in astrophysics [35].MMA processes can also be used, under suitable conditions, as building blocks for more complex models. We study in this paper the class of MMA stochastic volatility models. An example of the class is the supOU SV model, defined in [10,11], where the log-price process (of some financial asset) is defined for t ∈ R + as

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Section: R)-measurable Function With Values Inmentioning

confidence: 99%

Weak dependence and GMM estimation of supOU and mixed moving average processes

Curato¹,

Stelzer²

2019

Electron. J. Statist.

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“…It holds true that E 4 i=1 R d f i (t)dL(t) = (η − 3)σ 4 R d f 1 (u)f 2 (u)f 3 (u)f 4 (u)λ d (du) + σ d i=1,2 f i (u)λ d (du) R d i=3,4 f i (u)λ d (du) + σ d i=1,3 f i (u)λ d (du) R d i=2,4 f i (u)λ d (du) + σ i (u)λ d (du) R d i=2,3 f i (u)λ d (du).Proof. Follows directly from the proof of[3, Lemma 4.1]. Under the assumptions of Theorem 3.8, for ∆ p , ∆ q ∈ Z d , we have|Γ n |cov (γ * n (∆ p ), γ * n (∆ q )) → l∈Z d a l T l for n → ∞,whereT l :=(η − 3)σ 4 R d…”

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confidence: 89%

“…In [7], the asymptotics of the sample mean and sample autocovariance are studied when f decays sufficiently fast and L has finite second or fourth moment, respectively. [22] studies the situation when f decays slowly leading to a long-memory process X, while [11] considers the heavy tailed situation when the Lévy process L is in the domain of attraction of a stable non-normal distribution, and in [3] the case of random sampling when the process X is sampled at a renewal sequence is treated. Observe that all these results are in dimension d = 1 only.…”

Section: Introductionmentioning

confidence: 99%

Central Limit Theorems for Moving Average Random Fields with Non-Random and Random Sampling On Lattices

Berger

2019

Preprint

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For a Lévy basis L on R d and a suitable kernel function f : R d → R, consider the continuous spatial moving average field X = (X t ) t∈R d defined byBased on observations on finite subsets Γ n of Z d , we obtain central limit theorems for the sample mean and the sample autocovariance function of this process. We allow sequences (Γ n ) of deterministic subsets of Z d and of random subsets of Z d . The results generalise existing results for time indexed stochastic processes (i.e. d = 1) to random fields with arbitrary spatial dimension d, and additionally allow for random sampling. The results are applied to obtain a consistent and asymptotically normal estimator of µ > 0 in the stochastic partial differential equation (µ − ∆)X = dL in dimension 3, where L is Lévy noise.

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“…As was the case in Theorem 1.1, statement (i) is more general than statement (ii) of Theorem 1.2, but the latter may be convenient as it gives conditions on the decay rate of ϕ 1 and ϕ 2 at infinity. In relation to Theorem 1.2, it should be mentioned that limit theorems for the sample autocovariances of moving average processes (1.5) have been studied in [5,10,25].…”

Section: Introductionmentioning

confidence: 99%

Limit theorems for quadratic forms and related quantities of discretely sampled continuous-time moving averages

Nielsen

Pedersen

2019

ESAIM: PS

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The limiting behavior of Toeplitz type quadratic forms of stationary processes has received much attention through decades, particularly due to its importance in statistical estimation of the spectrum. In the present paper we study such quantities in the case where the stationary process is a discretely sampled continuous-time moving average driven by a Lévy process. We obtain sufficient conditions, in terms of the kernel of the moving and the coefficients of the quadratic form, ensuring that the centered and adequately normalized version of the quadratic form converges weakly to a Gaussian limit.

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On the sample autocovariance of a Lévy driven moving average process when sampled at a renewal sequence

Cited by 3 publications

References 0 publications

Weak dependence and GMM estimation of supOU and mixed moving average processes

Weak dependence and GMM estimation of supOU and mixed moving average processes

Central Limit Theorems for Moving Average Random Fields with Non-Random and Random Sampling On Lattices

Limit theorems for quadratic forms and related quantities of discretely sampled continuous-time moving averages

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