Abstract
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 $\rho$-mixing sequences, functions of linear processes with non-stationary innovations, locally stationary processes, 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.
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