Bounded law of the iterated logarithms for stationary random fields.

We give sufficient conditions for the bounded law of the iterated logarithms for strictly stationary random fields when the summation is done on rectangle. The study is done by the control of an appropriated maximal function. The case of functional of i.i.d. random fields, martingales with respect to the lexicographic order and orthomartingale is treated. Then results on projective conditions are derived. Applications to linear and Volterra random fields are given.