On non-standard limits of Brownian semi-stationary processes

TL;DR

This paper investigates how singularities in the weight function of Brownian semi-stationary processes cause non-standard asymptotic limits in high-frequency quadratic variation, revealing new limit behaviors and stable CLTs.

Contribution

It introduces novel asymptotic results for BSS processes with singularities, including non-standard limits and stable central limit theorems, enhancing understanding of their high-frequency behavior.

Findings

Singularities lead to non-standard limits of quadratic variation.

The limiting process is a convex combination of shifted integrals.

Stable central limit theorem is established for these limits.

Abstract

In this paper we present some new asymptotic results for high frequency statistics of Brownian semi-stationary processes. More precisely, we will show that singularities in the weight function, which is one of the ingredients of a BSS process, may lead to non-standard limits of the realised quadratic variation. In this case the limiting process is a convex combination of shifted integrals of the intermittency function. Furthermore, we will demonstrate the corresponding stable central limit theorem. Finally, we apply the probabilistic theory to study the asymptotic properties of the realised ratio statistics, which estimates the smoothness parameter of a BSS process.