  Many time series exhibit changes both in level and in variability. Generally,
it is more important to detect a change in the level, and changing or smoothly
evolving variability can confound existing tests. This paper develops a
framework for testing for shifts in the level of a series which accommodates
the possibility of changing variability. The resulting tests are robust both to
heteroskedasticity and serial dependence. They rely on a new functional central
limit theorem for dependent random variables whose variance can change or trend
in a substantial way. This new result is of independent interest as it can be
applied in many inferential contexts applicable to time series. Its application
to change point tests relies on a new approach which utilizes
Karhunen--Lo{\'e}ve expansions of the limit Gaussian processes. After
presenting the theory in the most commonly encountered setting of the detection
of a change point in the mean, we show how it can be extended to linear and
nonlinear regression. Finite sample performance is examined by means of a
simulation study and an application to yields on US treasury bonds.
