Wavelet-type expansion of the Rosenblatt process
The Rosenblatt process \(Z_{\kappa} = \{Z_{\kappa}(t), t \in {\mathbb R}\}\) (\(\kappa \in (1/4, 1/2)\)) is defined by \[ Z_{\kappa}(t) = K_{\kappa} \int'_{{\mathbb R}^2} \bigg\{\int_0^t (s - \xi_1)_+^{\kappa - 1} (s - \xi_2)_+^{\kappa - 1}ds\bigg\} \,dB(\xi_1) d B(\xi_2), \] where \(K_{\kappa}\) is a normalizing constant, \(\int'_{{\mathbb R}^2}\) denotes integration over \({\mathbb R}^2\) excluding the diagonal, \(x_+ = \max\{0, x\}\) and \(\{B(\xi), \xi \in {\mathbb R}\}\) is a standard Brownian motion. The process \(Z_{\kappa}\) is a non-Gaussian, \((2\kappa)\)-self-similar process with stationary increments. It can be realized as the weak limit of normalized sums of certain stationary non-Gaussian random variables with long-range dependence [\textit{M. S. Taqqu}, Z. Wahrscheinlichkeitstheorie Verw. Geb. 31, 287--302 (1975; Zbl 0303.60033)]. Similar to fractional Brownian motion, the Rosenblatt process \(Z_{\kappa}\) is of importance both in theory and in applications. The author shows that the Rosenblatt process \(Z_{\kappa}\) admits a wavelet-type expansion which is almost surely convergent uniformly on compact intervals. This provides meaningful approximations of \(Z_{\kappa}\) and can also be used in fast simulations of the process.
scientific article; zbMATH DE number 2166810
scientific article