Weak and strong law of large numbers for strictly stationary Banach-valued random fields

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 [Formula: see text] and the convergence in [Formula: see text] of partials sums over squares or rectangles of [Formula: see text]. Approximation by multi-indexed martingales as well as by [Formula: see text]-dependent random fields are investigated. Applications to functions of [Formula: see text]-independent Bernoulli shifts and to functionals of i.i.d. random fields are also provided.