We establish new exponential inequalities for partial sums of random fields. Next, using classicalchaining arguments, we give sufficient conditions for partial sum processes indexed by large classes ofsets to converge to a set-indexed Brownian motion. For stationary fields of bounded random variables, thecondition is expressed in terms of a series of conditional expectations. For non-uniform ϕ-mixingrandom fields, we require both finite fourth moments and an algebraic decay of the mixing coefficients.
