On a linear functional for infinitely divisible moving average random fields
Open Access
- 22 October 2019
- journal article
- research article
- Published by VTeX in Modern Stochastics: Theory and Applications
- Vol. 6 (4), 1-36
- https://doi.org/10.15559/19-vmsta143
Abstract
Publisher: VTeX - Solutions for Science Publishing, Journal: Modern Stochastics - Theory and Applications, Title: On a linear functional for infinitely divisible moving average random fields, Authors: Stefan Roth , Given a low-frequency sample of the infinitely divisible moving average random field $\{{\textstyle\int _{{\mathbb{R}^{d}}}}f(t-x)\Lambda (dx),\hspace{2.5pt}t\in {\mathbb{R}^{d}}\}$, in [13] we proposed an estimator $\widehat{u{v_{0}}}$ for the function $\mathbb{R}\ni x\mapsto u(x){v_{0}}(x)=(u{v_{0}})(x)$, with $u(x)=x$ and ${v_{0}}$ being the Lévy density of the integrator random measure Λ. In this paper, we study asymptotic properties of the linear functional ${L^{2}}(\mathbb{R})\ni v\mapsto {\left\langle v,\widehat{u{v_{0}}}\right\rangle _{{L^{2}}(\mathbb{R})}}$, if the (known) kernel function f has a compact support. We provide conditions that ensure consistency (in mean) and prove a central limit theorem for it.
Keywords
This publication has 17 references indexed in Scilit:
- A Donsker theorem for Lévy measuresJournal of Functional Analysis, 2012
- Stationary infinitely divisible processesBrazilian Journal of Probability and Statistics, 2011
- Nonparametric estimation for pure jump Lévy processes based on high frequency dataStochastic Processes and their Applications, 2009
- Nonparametric estimation for Lévy processes from low-frequency observationsBernoulli, 2009
- Lévy-based growth modelsBernoulli, 2008
- Ambit Processes; with Applications to Turbulence and Tumour GrowthPublished by Springer Science and Business Media LLC ,2007
- Normal approximation under local dependenceThe Annals of Probability, 2004
- Exponential inequalities and functional central limit theorems for random fieldsESAIM: Probability and Statistics, 2001
- Some Bounds of Cumulants of m‐Dependent Random FieldsMathematische Nachrichten, 1990
- Spectral representations of infinitely divisible processesProbability Theory and Related Fields, 1989