  Consider a Bernoulli random field satisfying the Hannan's condition.
Recently, invariance principles for partial sums of random fields over
rectangular index sets are established. In this note we complement previous
results by investigating limit theorems for weighted Bernoulli random fields,
including central limit theorems for partial sums over arbitrary index sets and
invariance principles for Gaussian random fields. Most results improve earlier
ones on Bernoulli random fields under Wu's condition, which is stronger than
Hannan's condition.
