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

Suppose a Poisson process is observed on the unit interval. The scan statistic is defined as the maximum number of events observed as a window of fixed width is moved across the interval, and the distribution under homogeneity has been widely studied. Frequently, we may not wish to specify the window width in advance but to consider scan statistics with varying window widths. We propose a modification of the scan statistic based on a likelihood ratio criterion. This leads to a boundary-crossing problem for a two-dimensional random field, which we approximate using a large-deviation scaling under homogeneity. Similar results are obtained for Poisson processes observed in two dimensions. Numerical computations and simulations are used to illustrate the accuracy of the approximations.