Non-central limit theorem for the oscillation of the Rosenblatt process

Let (Z_t^H, t ⩾ 0) be the Rosenblatt process with Hurst index H∈(12,1). 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 MQ(Xε)=∫01Q(Xtε)dt converges as ε → 0, almost surely and in L^q(Ω) for any q ⩾ 1, to EQ(Z₁^H) for any polynomial function Q with EQ(Z₁^H) < ∞. We also obtain a second order result, i.e., after a proper renormalization, the quantity M_Q(X^ε) − EQ(Z₁^H) converges as ε → 0, almost surely and in L^q(Ω) for any q ⩾ 1, to a Rosenblatt-distributed random variable.