Weak and strong 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.