On the Hsu–Robbins–Erdős–Spitzer–Baum–Katz theorem for random fields

The by now classical results on convergence rates in the law of large numbers involving the sums ∑ n = 1 ∞ n α r − 2 P ( | S n | > n α ε ) , where r > 0 , α > 1 / 2 , such that α r ⩾ 1 has been extended to the case α = 1 / 2 by adding additional logarithms. All of this has been generalized to random fields by the first named author in [A. Gut, Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices, Ann. Probab. 6 (1978) 469–482; A. Gut, Convergence rates for probabilities of moderate deviations for sums of random variables with multidimensional indices, Ann. Probab. 8 (1980) 298–313]. The purpose of the present paper is to treat the case when the αʼs differ in the different directions of the field, as well as mixed cases with some αʼs equal to 1/2 with added logarithms and/or iterated ones.