## Electronic Journal of Probability

### Weighted power variations of iterated Brownian motion

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

#### Article information

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

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

https://projecteuclid.org/euclid.ejp/1464819116

Digital Object Identifier
doi:10.1214/EJP.v13-534

Mathematical Reviews number (MathSciNet)
MR2430706

Zentralblatt MATH identifier
1193.60028

Rights

#### Citation

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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