Weak and strong law of large numbers for strictly stationary Banach-valued random fields: Law of large numbers for strictly stationary Banach-valued random fields
Abstract
In this paper, we investigate the law of large numbers for strictly stationary random fields, that is, we provide sufficient conditions on the moments and the dependence of the random field in order to guarantee the almost sure convergence to $0$ and the convergence in $\mathbb L^p$ of partials sums over squares or rectangles of $\mathbb Z^d$. Approximation by multi-indexed martingales as well as by $m$-dependent random fields are investigated. Applications to functions of $d$-independent Bernoulli shifts and to functionals of i.i.d.\ random fields are also provided.
Fields of Study & Subjects
Laser Design and Applications
Electron and X-Ray Spectroscopy Techniques
Geophysics and Sensor Technology
Publication details
- Journal / source
- HAL (Le Centre pour la Communication Scientifique Directe)
- Publication date
- 29 April 2026
- Language
- EN
- Document type
- Preprint