  In this paper, we develop necessary and sufficient conditions for the
validity of a martingale approximation for the partial sums of a stationary
process in terms of the maximum of consecutive errors. Such an approximation is
useful for transferring the conditional functional central limit theorem from
the martingale to the original process. The condition found is simple and well
adapted to a variety of examples, leading to a better understanding of the
structure of several stochastic processes and their asymptotic behaviors. The
approximation brings together many disparate examples in probability theory. It
is valid for classes of variables defined by familiar projection conditions
such as the Maxwell--Woodroofe condition, various classes of mixing processes,
including the large class of strongly mixing processes, and for additive
functionals of Markov chains with normal or symmetric Markov operators.
