We discuss the problem of positive-definite completion: extending a partially specified covariance kernel from a subdomain Ω of the unit square [0,1]^2 to a covariance kernel on the entire unit square [0,1]^2. For a broad class of subdomains Ω, we obtain a complete picture. Namely, we demonstrate that a canonical extension always exists and can be explicitly constructed. We characterise all possible extensions as suitable perturbations of the canonical extension, and determine necessary and sufficient conditions for a unique extension to exist. We then re-interpret the canonical extension as a graphical model on the associated Gaussian process. We show that this leads to a valid and operational definition for arbitrarily indexed (e.g. uncountably infinitely) Gaussian processes, based directly on the covariance kernel, and describe how this allows for nonparametric estimation of the underlying Markov structure. Based on joint work in collaboration with K.G. Waghmare (ETH Zürich).

Short Bio:
Victor M. Panaretos is Full Professor and former Director at the Mathematics Institute at the EPFL. He received his PhD in 2007 from UC Berkeley, advised by David Brillinger. Upon graduation he was appointed as Assistant Professor at the EPFL, where he rose the ranks to Full Professor. He received the Erich Lehmann Award for an Outstanding PhD (UC Berkeley, 2007), an ERC Starting Grant Award (2011) and was named "One of 40 extraordinary scientists under 40" by the World Economic Forum (2014). He is an Elected Member of the ISI (2008) and a Fellow of the IMS (2019). He was the Bernoulli Society Forum Lecturer in 2019, and will be an IMS Medallion Lecturer in 2025. He is an Associate Editor for Biometrika, and the Journal of the American Statistical Association (Theory and Methods), and previously for the Annals of Statistics, Annals of Applied Statistics, and Electronic Journal of Statistics. He has served the discipline from various posts, most notably as President of the Bernoulli Society.