## Bernoulli

• Bernoulli
• Volume 17, Number 4 (2011), 1159-1194.

### Multipower variation for Brownian semistationary processes

#### Abstract

In this paper we study the asymptotic behaviour of power and multipower variations of processes $Y$: $$Y_t = ∫_{−∞}^tg(t − s)σ_sW(\mathrm{d}s) + Z_t,$$ where $g : (0, ∞) → ℝ$ is deterministic, $σ > 0$ 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 $σ$. 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 $σ$. 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.

#### Article information

Source
Bernoulli, Volume 17, Number 4 (2011), 1159-1194.

Dates
First available in Project Euclid: 4 November 2011

Permanent link to this document
https://projecteuclid.org/euclid.bj/1320417500

Digital Object Identifier
doi:10.3150/10-BEJ316

Mathematical Reviews number (MathSciNet)
MR2854768

Zentralblatt MATH identifier
1244.60039

#### Citation

Barndorff-Nielsen, Ole E.; Corcuera, José Manuel; Podolskij, Mark. Multipower variation for Brownian semistationary processes. Bernoulli 17 (2011), no. 4, 1159--1194. doi:10.3150/10-BEJ316. https://projecteuclid.org/euclid.bj/1320417500

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