Bernoulli

  • Bernoulli
  • Volume 24, Number 4A (2018), 3117-3146.

On limit theory for Lévy semi-stationary processes

Andreas Basse-O’Connor, Claudio Heinrich, and Mark Podolskij

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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.

Article information

Source
Bernoulli, Volume 24, Number 4A (2018), 3117-3146.

Dates
Received: April 2016
Revised: February 2017
First available in Project Euclid: 26 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.bj/1522051235

Digital Object Identifier
doi:10.3150/17-BEJ956

Mathematical Reviews number (MathSciNet)
MR3779712

Zentralblatt MATH identifier
06853275

Keywords
high frequency data Lévy semi-stationary processes limit theorems power variation stable convergence

Citation

Basse-O’Connor, Andreas; Heinrich, Claudio; Podolskij, Mark. On limit theory for Lévy semi-stationary processes. Bernoulli 24 (2018), no. 4A, 3117--3146. doi:10.3150/17-BEJ956. https://projecteuclid.org/euclid.bj/1522051235


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