  Let $X_1, X_2,\dots$ be a short-memory linear process of random variables.
For $1\leq q<2$, let $\cF$ be a bounded set of real-valued functions on $[0,1]$
with finite $q$-variation. It is proved that
$\{n^{-1/2}\sum_{i=1}^nX_if(i/n)\colon\,f\in\cF\}$ converges in outer
distribution in the Banach space of bounded functions on $\cF$ as $n\to\infty$.
Several applications to a regression model and a multiple change point model
are given.
