Exact strong laws of large numbers for independent random fields

Let $\{ X_{\underline{n}}, \underline{n}\in \mathbb{N}^{d}\}$ be a family of independent random variables with multidimensional indices (a random field) with the same distribution as the r.v. $X.$ A necessary and sufficient condition for the strong law of large numbers in this setting is $\mathbb E \vert X\vert \log _{+}^{d-1}\vert X\vert \lt \infty.$ Our goal is to study the almost sure convergence of normalized or weighted sums in the case when this moment condition is not satisfied.