Article
| Title: | Non-central limit theorem for the oscillation of the Rosenblatt process (English) |
| Author: | Araya, Héctor |
| Author: | Tudor, Ciprian A. |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 75 |
| Issue: | 4 |
| Year: | 2025 |
| Pages: | 1255-1274 |
| Summary lang: | English |
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| Category: | math |
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| 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. (English) |
| Keyword: | multiple stochastic integral |
| Keyword: | Wiener chaos |
| Keyword: | Rosenblatt process |
| Keyword: | oscillation of stochastic process |
| MSC: | 60F05 |
| MSC: | 60G15 |
| MSC: | 60H05 |
| MSC: | 60H07 |
| DOI: | 10.21136/CMJ.2025.0040-25 |
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| Date available: | 2025-12-20T07:23:50Z |
| Last updated: | 2025-12-22 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153241 |
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