Behavior of the Hermite sheet with respect to the Hurst index
Hermite processes are \(d\)-parameter fractional sheets with multidimensional Hurst index \(H = (H_1 , \ldots , H_d ) \in (1/2,1)^d\), which take values in the \(q\)-th Wiener chaos. They generalize the construction of fractional Brownian motion (fBm) and of the Rosenblatt process, which take values in the first and second Wiener chaos, respectively. In this paper, the weak convergence in \(C([0,T]^d)\) of Hermite processes is characterized from the convergence to \(1/2\) or to \(1\) of the components of the index \(H = (H_1 , \ldots , H_d )\). In particular, the convergence to a \(d\)-parameter centered Gaussian process with known covariance is shown when at least one component of \(H = (H_1 , \ldots , H_d )\) tends to \(1/2\). On the other hand, when at least one component of \(H = (H_1 , \ldots , H_d )\) tends to \(1\) and none of the remaining components converges to \(1/2\), the limit is non-Gaussian and described explicitly as a \(q\)-th Wiener chaos process. The proofs use the fourth moment theorem and non-central approximations of the Hermite sheet.
scientific article; zbMATH DE number 7074620
scientific article