Let (Z t H , t ⩾ 0) be the Rosenblatt process with Hurst index \(H\in({{1}\over{2}},1)\) . We analyze the limit behavior of the oscillation of the Rosenblatt process given by X t H,ε = ε−H (Z t+ε H − Z t H ) with ε > 0 and t ⩾ 0. Based on the Wiener chaos expansion, we prove that the quantity \(M_{Q} (X^{\varepsilon})= \int_{0}^{1} Q (X ^{\varepsilon}_{t}) \,{\rm d}t\) converges as ε → 0, almost surely and in Lq(Ω) for any q ⩾ 1, to EQ(Z 1 H ) for any polynomial function Q with EQ(Z 1 H ) < ∞. We also obtain a second order result, i.e., after a proper renormalization, the quantity MQ(Xε) − EQ(Z 1 H ) converges as ε → 0, almost surely and in Lq(Ω) for any q ⩾ 1, to a Rosenblatt-distributed random variable.
