Multi-dimensional normal approximation of heavy-tailed moving averages

Azmoodeh, Ehsan; Ljungdahl, Mathias Mørck; Thäle, Christoph

doi:10.1016/j.spa.2021.11.011

Cited by 2 publications

(5 citation statements)

References 11 publications

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“…More specifically, we shall assume the existence of a constant K > 0 together with powers

α > 0

and

κ \in ℝ

for which it holds

| g (x) | \leq K (x^{κ} 1_{[0, 1)} (x) + x^{- α} 1_{[1, \infty)} (x)) for all x \in ℝ .

We are interested in (scaled) partial sums of multivariate functionals of the vectors (( X s +1 , … , X s + m )) s ≥ 0 :

V_{n} (X; f) = \frac{1}{\sqrt{n}} \sum_{s = 0}^{n prefix- m} (f (X_{s + 1}, \dots, X_{s + m}) - 𝔼 [f (X_{1}, \dots, X_{m})]),

where

f : ℝ^{m} \to ℝ^{d}

is a suitable Borel function. Adhering to Azmoodeh et al (2020, remark 2.4(iii)) the following result holds. Below

C_{b}^{2} (ℝ^{m}, ℝ^{d})

denotes the space of twice differentiable functions

f : ℝ^{m} \to ℝ^{d}

such that f and all of its first and second‐order derivatives are bounded and ...…”

Section: Introductionsupporting

confidence: 63%

“…Remark If we drop the requirement for estimation of

β

we can consider a larger class of Lévy drivers. Indeed, according to Azmoodeh et al (2020) the statement of Theorem 1 still holds for a symmetric Lévy process L , which admits a Lévy density

ν

such that

ν (x) \leq C | x |^{- 1 - β} for all x \neq 0 .

In this case the characteristic function takes on a more complicated form. Indeed, by Rajput and Rosinski (1989, theorem 2.7) it holds that

𝔼 [e^{i {⟨ u, (X_{1}, \dots, X_{m}) ⟩}_{ℝ^{m}}}] = \exp (\int_{ℝ} \int_{ℝ} [\cos ({⟨ u, x {(g_{ξ} (z + i))}_{i = 0, \dots, m - 1} ⟩}_{ℝ^{m}}) - 1] ν (d x) d z) .

In principle, the asymptotic theory of Theorem 2 can be extended to this more general setting.…”

Section: The Setting and Main Resultsmentioning

confidence: 94%

“…If we drop the requirement for estimation of 𝛽 we can consider a larger class of Lévy drivers. Indeed, according to Azmoodeh et al (2020) the statement of Theorem 1 still holds for a symmetric Lévy process L, which admits a Lévy density 𝜈 such that…”

Section: Parametric Estimation Via Minimal Contrast Approachmentioning

confidence: 99%

“…The aim of this article is to construct estimators of the kernel function and the stability index in the general setting of a multidimensional parameter space. Instead of relying on existing theory Basse‐O'Connor et al (2017, 2019) and Pipiras and Taqqu (2003), which only accounts for the marginal law of the underlying model, we shall use the framework from the recent paper Azmoodeh et al (2020), which is tailor‐made for the study of Gaussian fluctuations of functionals of multiple heavy‐tailed moving averages, to estimate the multidimensional parameter. While our approach is similar in spirit to the univariate framework of Ljungdahl and Podolskij (2019, 2020) from the theoretical viewpoint, there are some important differences.…”

Section: Introductionmentioning

confidence: 99%

“…The paper Azmoodeh et al (2020) additionally provides Berry–Esseen‐type bounds for an appropriate distance between probability laws on

ℝ^{d}

, but Theorem 1 is sufficient for our statistical analysis. We remark that the limit theory for bounded f in the case of m = 1 and general

d \in ℕ

is handled in Pipiras et al (2007), but it is actually the reverse situation, that is,

m \in ℕ

and d = 1, that is the most needed.…”

Section: Introductionmentioning

confidence: 99%

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Multidimensional parameter estimation of heavy‐tailed moving averages

Ljungdahl

Podolskij

2021

Scandinavian J Statistics

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In this article we present a parametric estimation method for certain multiparameter heavy-tailed Lévy-driven moving averages. The theory relies on recent multivariate central limit theorems obtained via Malliavin calculus on Poisson spaces. Our minimal contrast approach is related to previous papers, which propose to use the marginal empirical characteristic function to estimate the one-dimensional parameter of the kernel function and the stability index of the driving Lévy motion. We extend their work to allow for a multiparametric framework that in particular includes the important examples of the linear fractional stable motion, the stable Ornstein-Uhlenbeck process, certain CARMA(2, 1) models, and Ornstein-Uhlenbeck processes with a periodic component among other models. We present both the consistency and the associated central limit theorem of the minimal contrast estimator. Furthermore, we demonstrate numerical analysis to uncover the finite sample performance of our method.

show abstract

“…More specifically, we shall assume the existence of a constant K > 0 together with powers

α > 0

and

κ \in ℝ

for which it holds

| g (x) | \leq K (x^{κ} 1_{[0, 1)} (x) + x^{- α} 1_{[1, \infty)} (x)) for all x \in ℝ .

We are interested in (scaled) partial sums of multivariate functionals of the vectors (( X s +1 , … , X s + m )) s ≥ 0 :

V_{n} (X; f) = \frac{1}{\sqrt{n}} \sum_{s = 0}^{n prefix- m} (f (X_{s + 1}, \dots, X_{s + m}) - 𝔼 [f (X_{1}, \dots, X_{m})]),

where

f : ℝ^{m} \to ℝ^{d}

is a suitable Borel function. Adhering to Azmoodeh et al (2020, remark 2.4(iii)) the following result holds. Below

C_{b}^{2} (ℝ^{m}, ℝ^{d})

denotes the space of twice differentiable functions

f : ℝ^{m} \to ℝ^{d}

such that f and all of its first and second‐order derivatives are bounded and ...…”

Section: Introductionsupporting

confidence: 63%

“…Remark If we drop the requirement for estimation of

β

we can consider a larger class of Lévy drivers. Indeed, according to Azmoodeh et al (2020) the statement of Theorem 1 still holds for a symmetric Lévy process L , which admits a Lévy density