Electronic Journal of Probability

Weighted power variations of iterated Brownian motion

Ivan Nourdin and Giovanni Peccati

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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 Réveillac (2008), concerning the weighted power variations of fractional Brownian motion with Hurst index $H=1/4$.

Article information

Electron. J. Probab., Volume 13 (2008), paper no. 43, 1229-1256.

Accepted: 3 August 2008
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60G18: Self-similar processes 60K37: Processes in random environments

Brownian motion Brownian motion in random scenery Iterated Brownian motion Limit theorems Weighted power variations

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Nourdin, Ivan; Peccati, Giovanni. Weighted power variations of iterated Brownian motion. Electron. J. Probab. 13 (2008), paper no. 43, 1229--1256. doi:10.1214/EJP.v13-534. https://projecteuclid.org/euclid.ejp/1464819116

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