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 $H=1/4$.
Highlights
Introduction and main resultsThe characterization of the single-path behaviour of a given stochastic process is often based on the study of its power variations
The aim of this paper is to study the asymptotic behaviour, for every integer κ 2 and for n → ∞, of the κ-power variations associated with a remarkable non-Gaussian and selfsimilar process with stationary increments, known as iterated Brownian motion
We focus on the iterated Brownian motion, which is a continuous non
Summary
Introduction and main resultsThe characterization of the single-path behaviour of a given stochastic process is often based on the study of its power variations. The aim of this paper is to study the asymptotic behaviour, for every integer κ 2 and for n → ∞, of the (dyadic) κ-power variations associated with a remarkable non-Gaussian and selfsimilar process with stationary increments, known as iterated Brownian motion Our main result is the following: Theorem 1.2 Let f : R → R belong to C2 with f ′ and f ′′ bounded, and κ 2 be an integer.
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