Let 5e03aa5d5099f426aeb1" title="Click to view the MathML source">0<α≤2. Let 5e9a3e25627eb6032" title="Click to view the MathML source">Nd be the 5e4793256e93ec8c12a4942" title="Click to view the MathML source">d-dimensional lattice equipped with the coordinate-wise partial order 5e446f3bb8a4a1a9b1dc3adda584dd0" title="Click to view the MathML source">≤, where d≥1 is a fixed integer. For , define 5e32a33c7f9ce56ca8e42">. Let e15cbef84622bfde111c8538baa9">05" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0167715216301791-si8.gif"> be a field of independent and identically distributed real-valued random variables. Set , and write . This note is devoted to an extension of a strong limit theorem of Mikosch (1984). By applying an idea of Li and Chen (2014) and the classical Marcinkiewicz–Zygmund strong law of large numbers for random fields, we obtain necessary and sufficient conditions for