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November 2018 On limit theory for Lévy semi-stationary processes
Andreas Basse-O’Connor, Claudio Heinrich, Mark Podolskij
Bernoulli 24(4A): 3117-3146 (November 2018). DOI: 10.3150/17-BEJ956

Abstract

In this paper, we present some limit theorems for power variation of Lévy semi-stationary processes in the setting of infill asymptotics. Lévy semi-stationary processes, which are a one-dimensional analogue of ambit fields, are moving average type processes with a multiplicative random component, which is usually referred to as volatility or intermittency. From the mathematical point of view this work extends the asymptotic theory investigated in (Power variation for a class of stationary increments Lévy driven moving averages. Preprint), where the authors derived the limit theory for $k$th order increments of stationary increments Lévy driven moving averages. The asymptotic results turn out to heavily depend on the interplay between the given order of the increments, the considered power $p>0$, the Blumenthal–Getoor index $\beta\in(0,2)$ of the driving pure jump Lévy process $L$ and the behaviour of the kernel function $g$ at $0$ determined by the power $\alpha$. In this paper, we will study the first order asymptotic theory for Lévy semi-stationary processes with a random volatility/intermittency component and present some statistical applications of the probabilistic results.

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Andreas Basse-O’Connor. Claudio Heinrich. Mark Podolskij. "On limit theory for Lévy semi-stationary processes." Bernoulli 24 (4A) 3117 - 3146, November 2018. https://doi.org/10.3150/17-BEJ956

Information

Received: 1 April 2016; Revised: 1 February 2017; Published: November 2018
First available in Project Euclid: 26 March 2018

zbMATH: 06853275
MathSciNet: MR3779712
Digital Object Identifier: 10.3150/17-BEJ956

Rights: Copyright © 2018 Bernoulli Society for Mathematical Statistics and Probability

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Vol.24 • No. 4A • November 2018
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