Scaling limits for sample autocovariance operators of Hilbert space-valued linear processes
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911015825833984 |
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| author | Düker, Marie-Christine Zoubouloglou, Pavlos |
| author_facet | Düker, Marie-Christine Zoubouloglou, Pavlos |
| contents | This article considers linear processes with values in a separable Hilbert space exhibiting long-range dependence. The scaling limits for the sample autocovariance operators at different time lags are investigated in the topology of their respective Hilbert spaces. Distinguishing two different regimes of long-range dependence, the limiting object is either a Hilbert space-valued Gaussian or a Hilbert space-valued non-Gaussian random variable. The latter can be represented as a unitary transformation of double Wiener-Itô integrals with sample paths in a function space. This work is the first to show weak convergence to such double stochastic integrals with sample paths in infinite dimensions. The result generalizes the well known convergence to a Hermite process in finite dimensions, introducing a new domain of attraction for probability measures in Hilbert spaces. The key technical contributions include the introduction of double Wiener-Itô integrals with values in a function space and with dependent integrators, as well as establishing sufficient conditions for their existence as limits of sample autocovariance operators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_17168 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Scaling limits for sample autocovariance operators of Hilbert space-valued linear processes Düker, Marie-Christine Zoubouloglou, Pavlos Probability Statistics Theory This article considers linear processes with values in a separable Hilbert space exhibiting long-range dependence. The scaling limits for the sample autocovariance operators at different time lags are investigated in the topology of their respective Hilbert spaces. Distinguishing two different regimes of long-range dependence, the limiting object is either a Hilbert space-valued Gaussian or a Hilbert space-valued non-Gaussian random variable. The latter can be represented as a unitary transformation of double Wiener-Itô integrals with sample paths in a function space. This work is the first to show weak convergence to such double stochastic integrals with sample paths in infinite dimensions. The result generalizes the well known convergence to a Hermite process in finite dimensions, introducing a new domain of attraction for probability measures in Hilbert spaces. The key technical contributions include the introduction of double Wiener-Itô integrals with values in a function space and with dependent integrators, as well as establishing sufficient conditions for their existence as limits of sample autocovariance operators. |
| title | Scaling limits for sample autocovariance operators of Hilbert space-valued linear processes |
| topic | Probability Statistics Theory |
| url | https://arxiv.org/abs/2506.17168 |