Strong limit theorems for random fields

The aim of the present paper is to review some joint work with Ulrich Stadtmuller concerning random field analogs of the classical strong laws. In the first half we start, as background information, by quoting the law of large numbers and the law of the iterated logarithm for random sequences as well as for random fields, and the law of the single logarithm for sequences. We close with a one-dimensional LSL pertaining to windows, whose edges expand in an "almost linear fashion", viz., the length of the nth window equals, for example, n/log n or n/log log n. A sketch of the proof will also be given. The second part contains some extensions of the LSL to random fields, after which we turn to convergence rates in the law of large numbers. Departing from the now legendary Baum-Katz theorem in 1965, we review a number of results in the multiindex setting. Throughout main emphasis is on the case of "non-equal expansion rates", viz., the case when the edges along the different directions expand at different rates. Some results when the power weights are replaced by almost exponential weights are also given. We close with some remarks on martingales and the strong law.