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