Weighted power variations of iterated Brownian motion

TL;DR

This paper analyzes the asymptotic behavior of weighted power variations in iterated Brownian motion, revealing their convergence properties and expressing limits through independent Brownian motions and local times, extending existing stochastic process theories.

Contribution

It introduces a comprehensive characterization of weighted power variations for iterated Brownian motion, expanding the understanding of their asymptotic distributions and connections to Brownian motion in random scenery.

Findings

Weak convergence of weighted power variations established

Limiting laws expressed via independent Brownian motions and local times

Extends previous results on fractional Brownian motion with H=1/4

Abstract

We characterize the asymptotic behaviour of the weighted power variation processes associated with iterated Brownian motion. We prove weak convergence results in the sense of finite dimensional distributions, and show that the laws of the limiting objects can always be expressed in terms of three independent Brownian motions X, Y and B, as well as of the local times of Y. In particular, our results involve ``weighted'' versions of Kesten and Spitzer's Brownian motion in random scenery. Our findings extend the theory initiated by Khoshnevisan and Lewis (1999), and should be compared with the recent result by Nourdin and R\'eveillac (2008), concerning the weighted power variations of fractional Brownian motion with Hurst index H=1/4.