On limit theory for Levy semi-stationary processes

TL;DR

This paper develops limit theorems for the power variation of Levy semi-stationary processes, extending previous work to include random volatility and exploring the influence of jump activity, increment order, and kernel behavior.

Contribution

It introduces the first order asymptotic theory for Levy semi-stationary processes with stochastic volatility, expanding the mathematical framework for these models.

Findings

Asymptotic results depend on increment order, power p, Blumenthal–Getoor index, and kernel behavior.

The work extends previous limit theorems to processes with random volatility.

Statistical applications demonstrate the practical relevance of the theoretical results.

Abstract

In this paper we present some limit theorems for power variation of L\'evy semi-stationary processes in the setting of infill asymptotics. L\'evy 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 [14], where the authors derived the limit theory for $k$th order increments of stationary increments L\'evy 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 L\'evy 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 L\'evy semi-stationary processes with a random volatility/intermittency component and present some statistical applications of the probabilistic results.