  We present a general functional central limit theorem started at a point also
known under the name of quenched. As a consequence, we point out several new
classes of stationary processes, defined via projection conditions, which
satisfy this type of asymptotic result. One of the theorems shows that if a
Markov chain is stationary ergodic and reversible, this result holds for
bounded additive functionals of the chain which have a martingale coboundary in
L_1 representation. Our results are also well adapted for strongly mixing
sequences providing for this case an alternative, shorter approach to some
recent results in the literature.
