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Keywords:
multiple stochastic integral; Wiener chaos; Rosenblatt process; oscillation of stochastic process
multiple stochastic integral; Wiener chaos; Rosenblatt process; oscillation of stochastic process
Summary:
Let $(Z ^{H}_{t}, t\geq 0)$ be the Rosenblatt process with Hurst index $H\in (\frac {1}{2},1 )$. We analyze the limit behavior of the oscillation of the Rosenblatt process given by $X^{H, \varepsilon }_{t} = \varepsilon ^{-H} (Z^{H}_{t+\varepsilon }- Z^{H}_{t})$ with $\varepsilon >0$ and $t\geq 0$. 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 $\varepsilon \to 0$, almost surely and in $ L^{q}(\Omega )$ for any $q\geq 1$, to $ {\bf E} Q(Z^{H}_{1})$ for any polynomial function $Q$ with $ {\bf E} Q(Z^{H}_{1})<\infty $. We also obtain a second order result, i.e., after a proper renormalization, the quantity $M_{Q} ( X ^{\varepsilon })-{\bf E} Q(Z^{H}_{1})$ converges as $\varepsilon \to 0$, almost surely and in $ L^{q}(\Omega )$ for any $q\geq 1$, to a Rosenblatt-distributed random variable.
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