Autoregressive Processes on Riemannian Manifolds

[2606.24771] Autoregressive Processes on Riemannian Manifolds Skip to main content Search Submit Donate Log in Mathematics > Statistics Theory arXiv:2606.24771 (math) [Submitted on 23 Jun 2026 ( v1 ), last revised 20 Sep 2026 (this version, v4)] Title: Autoregressive Processes on Riemannian Manifolds Authors: Meshal Abuqrais , Davide Pigoli View PDF HTML (experimental) Abstract: This paper introduces a Riemannian autoregressive (R-AR) model of order one for manifold-valued time series. The model is specified through an autoregressive process in the tangent space at a reference point, which is mapped to the manifold via the exponential map. It is characterised by two parameters: a reference point $\mu$, representing the intrinsic central tendency through the Fréchet mean, and an autoregressive parameter $\phi$, governing the dependence structure and stationarity properties. When these parameters are unknown, their estimation introduces geometric and probabilistic challenges arising from the intrinsic estimation of $\mu$ and from estimating $\phi$ as the base point varies over the manifold and the associated tangent spaces change. The estimation theory is developed using a strong law of large numbers for the sample Fréchet mean set of ergodic Markov chains, together with ergodic arguments for the strong consistency of the autoregressive parameter estimator. The framework is validated through numerical simulations in the hyperbolic plane and an application to aerosol size distributions on the Fisher-Rao manifold, demonstrating how the proposed model can characterise mean-reverting dynamics in nonlinear geometries. Subjects: Statistics Theory (math.ST) ; Probability (math.PR) Cite as: arXiv:2606.24771 [math.ST] (or arXiv:2606.24771v4 [math.ST] for this version) https://doi.org/10.48550/arXiv.2606.24771 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Meshal Abuqrais [ view email ] [v1] Tue, 23 Jun 2026 16:31:03 UTC (4,013 KB) [v2] Wed, 24 Jun 2026 03:07:12 UTC (4,013 KB) [v3] Tue, 25 Aug 2026 02:09:56 UTC (4,014 KB) [v4] Sun, 20 Sep 2026 02:07:08 UTC (4,014 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.ST < prev | next > new | recent | 2026-06 Change to browse by: math math.PR stat stat.TH References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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