Let 8cf5e03aa5d5099f426aeb1" title="Click to view the MathML source">0<α≤2. Let Nd be the 8c12a4942" title="Click to view the MathML source">d-dimensional lattice equipped with the coordinate-wise partial order 85e446f3bb8a4a1a9b1dc3adda584dd0" title="Click to view the MathML source">≤, where d≥1 is a fixed integer. For e51141d38df41ad5">, define 9cc51e9428c5e32a33c7f9ce56ca8e42">. Let 8538baa9"> be a field of independent and identically distributed real-valued random variables. Set , e708c87584f09a317df94154e80658"> 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