  We propose a family of CUSUM-based statistics to detect the presence of
changepoints in the deterministic part of the autoregressive parameter in a
Random Coefficient AutoRegressive (RCA) sequence. In order to ensure the
ability to detect breaks at sample endpoints, we thoroughly study weighted
CUSUM statistics, analysing the asymptotics for virtually all possible weighing
schemes, including the standardised CUSUM process (for which we derive a
Darling-Erdos theorem) and even heavier weights (studying the so-called R\'enyi
statistics). Our results are valid irrespective of whether the sequence is
stationary or not, and no prior knowledge of stationarity or lack thereof is
required. Technically, our results require strong approximations which, in the
nonstationary case, are entirely new. Similarly, we allow for
heteroskedasticity of unknown form in both the error term and in the stochastic
part of the autoregressive coefficient, proposing a family of test statistics
which are robust to heteroskedasticity, without requiring any prior knowledge
as to the presence or type thereof. Simulations show that our procedures work
very well in finite samples. We complement our theory with applications to
financial, economic and epidemiological time series.
