A Karhunen-Lo\`{e}ve Theorem for Random Flows in Hilbert spaces
Probability
2023-03-03 v2 Statistics Theory
Statistics Theory
Abstract
We develop a generalisation of Mercer's theorem to operator-valued kernels in infinite dimensional Hilbert spaces. We then apply our result to deduce a Karhunen-Lo\`eve theorem, valid for mean-square continuous Hilbertian functional data, i.e. flows in Hilbert spaces. That is, we prove a series expansion with uncorrelated coefficients for square-integrable random flows in a Hilbert space, that holds uniformly over time.
Cite
@article{arxiv.2303.00702,
title = {A Karhunen-Lo\`{e}ve Theorem for Random Flows in Hilbert spaces},
author = {Leonardo V. Santoro and Kartik G. Waghmare and Victor M. Panaretos},
journal= {arXiv preprint arXiv:2303.00702},
year = {2023}
}