{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,25]],"date-time":"2026-10-25T04:01:32Z","timestamp":1792900892844,"version":"4.5.0"},"reference-count":55,"publisher":"Institute of Mathematical Statistics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Ann. Statist."],"published-print":{"date-parts":[[2026,6,1]]},"abstract":"<jats:p>Gaussian Processes (GPs) are widely used to model dependencies in spatial statistics and machine learning. However, exact inference is computationally intractable for GP regression, with a time complexity of O(n3). The Vecchia approximation scales up computation by introducing sparsity into the spatial dependency structure, represented by a directed acyclic graph (DAG). Despite its practical popularity, this approach lacks rigorous theoretical foundations, and the choice of DAG structure remains an open problem. In this paper, we systematically study the Vecchia approximation of the popular, isotropic Mat\u00e9rn GP as standalone stochastic process and uncover key probabilistic and statistical properties. We propose selecting parent sets as norming sets with fixed cardinality in the Vecchia approximation. On the probabilistic side, we show that the conditional distributions of Mat\u00e9rn GPs, as well as their Vecchia approximations, can be characterized by polynomial interpolations. This enables us to establish several results on small ball probabilities and the Reproducing Kernel Hilbert Spaces (RKHSs) of Vecchia GPs. Building on these probabilistic results, we prove that in the nonparametric regression model, the corresponding posterior contracts around the truth at the optimal minimax rate, both under oracle rescaling and hierarchical tuning of the prior. We illustrate the theoretical findings through numerical experiments on synthetic datasets. Our core algorithms are implemented in C++ with an R interface.<\/jats:p>","DOI":"10.1214\/25-aos2604","type":"journal-article","created":{"date-parts":[[2026,6,25]],"date-time":"2026-06-25T13:49:41Z","timestamp":1782395381000},"page":"1289-1314","source":"Crossref","is-referenced-by-count":1,"title":["Vecchia Gaussian processes: On probabilistic and statistical properties"],"prefix":"10.1214","volume":"54","author":[{"given":"Botond","family":"Szabo","sequence":"first","affiliation":[{"name":"Bocconi Institute for Data Science and Analytics and Department of Decision Sciences, Bocconi University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4050-5874","authenticated-orcid":false,"given":"Yichen","family":"Zhu","sequence":"additional","affiliation":[{"name":"Department of Statistics and Actuarial Science, University of Hong Kong"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"48","doi-asserted-by":"publisher","unstructured":"<small>Szab\u00f3, B. T.<\/small>, <small>van der Vaart, A. W.<\/small> and <small>van Zanten, J. H.<\/small> (2013). Empirical Bayes scaling of Gaussian priors in the white noise model. <i>Electron. J. Stat.<\/i> <b>7<\/b> 991\u20131018.","DOI":"10.1214\/13-EJS798"},{"key":"25","doi-asserted-by":"publisher","unstructured":"<small>KAUFMAN, C. G.<\/small>, <small>SCHERVISH, M. J.<\/small> and <small>NYCHKA, D. W.<\/small> (2008). Covariance tapering for likelihood-based estimation in large spatial data sets. <i>J. Amer. Statist. Assoc.<\/i> <b>103<\/b> 1545\u20131555.","DOI":"10.1198\/016214508000000959"},{"key":"35","doi-asserted-by":"publisher","unstructured":"<small>PERUZZI, M.<\/small>, <small>BANERJEE, S.<\/small> and <small>FINLEY, A. O.<\/small> (2022). Highly scalable Bayesian geostatistical modeling via meshed Gaussian processes on partitioned domains. <i>J. Amer. Statist. 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Res.<\/i> <b>12<\/b> 2095\u20132119."},{"key":"9","doi-asserted-by":"publisher","unstructured":"<small>Cressie, N.<\/small> and <small>Johannesson, G.<\/small> (2008). Fixed rank kriging for very large spatial data sets. <i>J. R. Stat. Soc. Ser. B. Stat. Methodol.<\/i> <b>70<\/b> 209\u2013226.","DOI":"10.1111\/j.1467-9868.2007.00633.x"},{"key":"36","unstructured":"<small>Qui\u00f1onero-Candela, J.<\/small> and <small>Rasmussen, C. E.<\/small> (2005). A unifying view of sparse approximate Gaussian process regression. <i>J. Mach. Learn. Res.<\/i> <b>6<\/b> 1939\u20131959."},{"key":"10","unstructured":"<small>CRESSIE, N.<\/small> and <small>WIKLE, C. K.<\/small> (2011). <i>Statistics for Spatio-Temporal Data<\/i>. <i>Wiley Series in Probability and Statistics<\/i>. Wiley, Hoboken, NJ."},{"key":"11","doi-asserted-by":"publisher","unstructured":"<small>Datta, A.<\/small>, <small>Banerjee, S.<\/small>, <small>Finley, A. O.<\/small> and <small>Gelfand, A. E.<\/small> (2016). 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Available at arXiv:<a href=\"https:\/\/arxiv.org\/abs\/2401.15813\">2401.15813<\/a>."},{"key":"26","doi-asserted-by":"publisher","unstructured":"<small>KUELBS, J.<\/small>, <small>LI, W. V.<\/small> and <small>LINDE, W.<\/small> (1994). The Gaussian measure of shifted balls. <i>Probab. Theory Related Fields<\/i> <b>98<\/b> 143\u2013162.","DOI":"10.1007\/BF01192511"},{"key":"28","doi-asserted-by":"publisher","unstructured":"<small>LEDOUX, M.<\/small> (2006). Isoperimetry and Gaussian analysis. In <i>Lectures on Probability Theory and Statistics<\/i> (<i>Saint-Flour<\/i>, 1994). <i>Lecture Notes in Math.<\/i> <b>1648<\/b> 165\u2013294. Springer, Berlin.","DOI":"10.1007\/BFb0095676"},{"key":"29","unstructured":"<small>LEE, J.<\/small>, <small>BAHRI, Y.<\/small>, <small>NOVAK, R.<\/small>, <small>SCHOENHOLZ, S. S.<\/small>, <small>PENNINGTON, J.<\/small> and <small>SOHL-DICKSTEIN, J.<\/small> (2017). Deep neural networks as Gaussian processes. arXiv preprint. 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