Multi-Dimensional Normal Approximation of Heavy-Tailed Moving Averages

Abstract

In this paper we extend the refined second-order Poincaré inequality in [2] from a one-dimensional to a multi-dimensional setting. Its proof is based on a multivariate version of the Malliavin-Stein method for normal approximation on Poisson spaces. We also present an application to partial sums of vector-valued functionals of heavytailed moving averages. The extension we develop is not only in the co-domain of the functional, but also in its domain. Such a set-up has previously not been explored in the framework of stable moving average processes. It can potentially capture probabilistic properties which cannot be described solely by the one-dimensional marginals, but instead require the joint distribution.

Multi-dimensional parameter estimation of heavy-tailed moving averages

“…where f : R m → R d is a suitable Borel function. Adhering to [3,Remark 2.4(iii)] the following result holds. Below C 2 b (R m , R d ) denotes the space of twice differentiable functions f : R m → R d such that f and all of its first and second order derivatives are bounded and continuous.…”

“…If we drop the requirement for estimation of β we can consider a larger class of Lévy drivers. Indeed, according to [3] the statement of Theorem 1.1 still holds for a symmetric Lévy process L, which admits a Lévy density ν such that ν(x) ≤ C |x| −1−β for all x = 0.…”

“…The aim of this paper is to construct estimators of the kernel function and the stability index in the general setting of a multi-dimensional parameter space. Instead of relying on existing theory [4,5,18], which only accounts for the marginal law of the underlying model, we shall use the framework from the recent paper [3], which is tailor-made for the study of Gaussian fluctuations of functionals of multiple heavy-tailed moving averages, to estimate the multidimensional parameter.…”

“…The paper [3] additionally provides Berry-Esseen type bounds for an appropriate distance between probability laws on R d , but Theorem 1.1 is sufficient for our statistical analysis. We remark that the limit theory for bounded f in the case of m = 1 and general d ∈ N is handled in [19], but it is actually the reverse situation, i.e.…”

“…

In this paper we present a parametric estimation method for certain multi-parameter heavy-tailed Lévy-driven moving averages. The theory relies on recent multivariate central limit theorems obtained in [3] via Malliavin calculus on Poisson spaces. Our minimal contrast approach is related to the papers [15,14], 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.

…”