Multipower variation for Brownian semistationary processes

TL;DR

This paper investigates the asymptotic behavior of power and multipower variations of a class of stochastic processes modeling turbulence, providing new limit theorems applicable beyond semimartingale frameworks.

Contribution

It introduces a general central limit theorem for triangular Gaussian schemes and applies it to analyze multipower variations of non-semimartingale processes.

Findings

Derived limit theorems for multipower variations

Extended analysis to non-semimartingale processes

Applied results to turbulence data modeling

Abstract

In this paper we study the asymptotic behaviour of power and multipower variations of processes $Y$:\[Y_t=\int_{-\in fty}^tg(t-s)\sigma_sW(\mathrm{d}s)+Z_t,\] where $g:(0,\infty)\rightarrow\mathbb{R}$ is deterministic, $\sigma >0$ is a random process, $W$ is the stochastic Wiener measure and $Z$ is a stochastic process in the nature of a drift term. Processes of this type serve, in particular, to model data of velocity increments of a fluid in a turbulence regime with spot intermittency $\sigma$. The purpose of this paper is to determine the probabilistic limit behaviour of the (multi)power variations of $Y$ as a basis for studying properties of the intermittency process $\sigma$. Notably the processes $Y$ are in general not of the semimartingale kind and the established theory of multipower variation for semimartingales does not suffice for deriving the limit properties. As a key tool for the results, a general central limit theorem for triangular Gaussian schemes is formulated and proved. Examples and an application to the realised variance ratio are given.