Selfsimilar processes with stationary increments in the second Wiener chaos (Q2905818)

Stochastic processes belonging to the second Wiener chaos are given by double stochastic integrals over certain deterministic functions with respect to the Wiener process. For single stochastic integrals of this form, i.e., for the first Wiener chaos, it is known that the only self-similar finite variance processes with stationary increments are the fractional Brownian motions of Hurst parameter \(H\in(0,1]\). The authors study a two parameter generalization of the Rosenblatt process given by NEWLINE\[NEWLINEY_t=c\int_{\mathbb R^2}\left(\int_0^t(u-t_1)_+^{H_1/2-1}(u-t_2)_+^{H_2/2-1}\,du\right)\,dB_{t_1}\,dB_{t_2}NEWLINE\]NEWLINE with parameters \(H_1,H_2\in(0,1)\) such that \(H_1+H_2>1\). The normalizing constant \(c>0\) is chosen such that \(\operatorname{E}[Y_1^2]=1\). It is shown that these processes belong to the second Wiener chaos and are self-similar processes of Hurst index \(H=\frac12(H_1+H_2)\in(\frac12,1)\) with stationary increments. In contrast to the first Wiener chaos, the authors prove that the Hurst parameter does not uniquely determine the law of the process, which is now given by the cumulants functions rather than the covariance function of the process. For \(H>\frac34\), the processes \((Y_t)_{t\geq0}\) are further shown to appear as limits in non-central limit theorems (\(L^2\)-limit or limit for all finite-dimensional distributions) of certain partial sum processes involving two dependent fractional Brownian motions. The authors finally argue how their construction can be extended to find even more self-similar processes with stationary increments in the second Wiener chaos.