From intersection local time to the Rosenblatt process

Abstract

The Rosenblatt process was obtained by Taqqu (Z. Wahr. Verw. Geb. 31:287-302, 1975) from convergence in distribution of partial sums of strongly dependent random variables. In this paper, we give a particle picture approach to the Rosenblatt process with the help of intersection local time and white noise analysis, and discuss measuring its long-range dependence by means of a number called dependence exponent.

Behavior of the generalized Rosenblatt process at extreme critical exponent values

“…Recent studies on the Rosenblatt process Z γ (t) include Tudor and Viens [32], Bardet and Tudor [7], Arras [1], Maejima and Tudor [18], Veillette and Taqqu [33] and Bojdecki et al [9]. The Rosenblatt and the generalized Rosenblatt processes are of interest because they are the simplest extension to the non-Gaussian world of the Gaussian fractional Brownian motion.…”

Particle picture representation of the non-symmetric Rosenblatt process and Hermite processes of any order

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We provide a particle picture representation for the non-symmetric Rosenblatt process and for Hermite processes of any order, extending the result of Bojdecki, Gorostiza and Talarczyk in [3]. We show that these processes can be obtained as limits in the sense of finite-dimensional distributions of certain functionals of a system of particles evolving according to symmetric stable Lévy motions.

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“…The main scheme of the proofs of Theorems 1 and 2 is similar to the one employed in [3], in particular the idea of using Wick products of an appropriate S ′ -valued random variable. To stress some of the main differences and difficulties that had to be overcome in our case let us point out that in the case of Theorem 1 it was at first not at all clear what functional of a particle system can be used to approximate the non-symmetric Rosenblatt process.…”

“…The rest of the argument is exactly the same as in the proof of equation (6.26) in [3] because V δ f ǫ ∈ S(R) implies that Ψ f ǫ,δ,φ is in S(R 2 ). The convergence in (3.30) follows from Lemma 6.3 in [3]. For the proof of (3.31) we will look at Hermite processes from the point of view of multiple Wiener-Ito integrals.…”

“…Lemma 7 and Lemma 8.1 from [4] give us (3.28) (for details see the proof of equation (6.26) in [3]). The proof of (3.29) is very similar to the proof of equation (6.27) in [3] so we will only sketch it. Recalling (2.5) we may write…”