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Limit theorems for multivariate Brownian semistationary processes and feasible results

Part of: Stochastic processes Probability theory on algebraic and topological structures Limit theorems

Published online by Cambridge University Press:  03 September 2019

Riccardo Passeggeri  and
Almut E. D. Veraart
Riccardo Passeggeri*
Affiliation:
Imperial College London
Almut E. D. Veraart*
Affiliation:
Imperial College London
*
* Postal address: Department of Mathematics, Imperial College London, 180 Queen’s Gate, London, SW7 2AZ, UK.
* Postal address: Department of Mathematics, Imperial College London, 180 Queen’s Gate, London, SW7 2AZ, UK.
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Abstract

In this paper we introduce the multivariate Brownian semistationary (BSS) process and study the joint asymptotic behaviour of its realised covariation using in-fill asymptotics. First, we present a central limit theorem for general multivariate Gaussian processes with stationary increments, which are not necessarily semimartingales. Then, we show weak laws of large numbers, central limit theorems, and feasible results for BSS processes. An explicit example based on the so-called gamma kernels is also provided.

Keywords

Multivariate Brownian semistationary processcentral limit theoremlaw of large numbersfeasiblenonsemimartingaleWiener chaoshigh frequency dataintermittencygamma kernel

MSC classification

Primary: 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles 60G15: Gaussian processes 60B12: Limit theorems for vector-valued random variables (infinite-dimensional case)
Type
Original Article
Information
Advances in Applied Probability , Volume 51 , Issue 3 , September 2019 , pp. 667 - 716
Copyright
© Applied Probability Trust 2019 

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