An extension of a theorem of Mikosch

Let 0<α≤2. Let Nd be the d-dimensional lattice equipped with the coordinate-wise partial order ≤, where d≥1 is a fixed integer. For n=(n1,…,nd)∈Nd, define |n|=∏i=1dni. Let {X,Xn;n∈Nd} be a field of independent and identically distributed real-valued random variables. Set Sn=∑k≤nXk, n∈Nd and write logx=loge(e∨x),x≥0. 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 lim supn|Sn|(log|n|)−1=Δlimm→∞sup|n|≥m|Sn|(log|n|)−1=e1/αalmost surely .