  The results of Koml\'{o}s, Major and Tusn\'{a}dy give optimal Wiener
approximation of partial sums of i.i.d. random variables and provide an
extremely powerful tool in probability and statistical inference. Recently Wu
[Ann. Probab. 35 (2007) 2294--2320] obtained Wiener approximation of a class of
dependent stationary processes with finite $p$th moments, $2<p\le4$, with error
term $o(n^{1/p}(\log n)^{\gamma})$, $\gamma>0$, and Liu and Lin [Stochastic
Process. Appl. 119 (2009) 249--280] removed the logarithmic factor, reaching
the Koml\'{o}s--Major--Tusn\'{a}dy bound $o(n^{1/p})$. No similar results exist
for $p>4$, and in fact, no existing method for dependent approximation yields
an a.s. rate better than $o(n^{1/4})$. In this paper we show that allowing a
second Wiener component in the approximation, we can get rates near to
$o(n^{1/p})$ for arbitrary $p>2$. This extends the scope of applications of the
results essentially, as we illustrate it by proving new limit theorems for
increments of stochastic processes and statistical tests for short term
(epidemic) changes in stationary processes. Our method works under a general
weak dependence condition covering wide classes of linear and nonlinear time
series models and classical dynamical systems.
