First and Second Order Quadratic Variations. Wavelet-Type Variations

Abstract

In this chapter we study the asymptotic behavior of the quadratic variation (including standard quadratic, quadratic variation based on higher order increments or wavelet-type quadratic variations) for several self-similar processes, such as fractional Brownian motion, the Rosenblatt process, the Hermite process of general order or the solution to the linear heat equation. We prove Central or Non-Central Limit Theorems for the sequence of quadratic variations using chaos expansion into multiple Wiener-Ito integrals and Malliavin calculus.