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The celebrated results of Koml\'os, Major and Tusn\'ady [Z. Wahrsch. Verw.Gebiete 32 (1975) 111-131; Z. Wahrsch. Verw. Gebiete 34 (1976) 33-58] giveoptimal Wiener approximation for the partial sums of i.i.d. random variablesand provide a powerful tool in probability and statistics. In this paper weextend KMT approximation for a large class of dependent stationary processes,solving a long standing open problem in probability theory. Under the frameworkof stationary causal processes and functional dependence measures of Wu [Proc.Natl. Acad. Sci. USA 102 (2005) 14150-14154], we show that, under naturalmoment conditions, the partial sum processes can be approximated by Wienerprocess with an optimal rate. Our dependence conditions are mild and easilyverifiable. The results are applied to ergodic sums, as well as to nonlineartime series and Volterra processes, an important class of nonlinear processes.

Koml\'os-Major-Tusn\'ady approximation under dependence