  In this paper we develop non-stationary martingale techniques for dependent
data. We shall stress the non-stationary version of the projective
Maxwell-Woodroofe condition, which will be essential for obtaining maximal
inequalities and functional central limit theorem for the following examples:
nonstationary \r{ho}-mixing sequences, functions of linear processes with
non-stationary innovations, quenched version of the functional central limit
theorem for a stationary sequence, evolutions in random media such as a process
sampled by a shifted Markov chain.
