  In this paper, we introduce a new method for testing the stationarity of time
series, where the test statistic is obtained from measuring and maximising the
difference in the second-order structure over pairs of randomly drawn
intervals. The asymptotic normality of the test statistic is established for
both Gaussian and a range of non-Gaussian time series, and a bootstrap
procedure is proposed for estimating the variance of the main statistics.
Further, we show the consistency of our test under local alternatives. Due to
the flexibility inherent in the random, unsystematic sub-samples used for test
statistic construction, the proposed method is able to identify the intervals
of significant departure from the stationarity without any dyadic constraints,
which is an advantage over other tests employing systematic designs. We
demonstrate its good finite sample performance on both simulated and real data,
particularly in detecting localised departure from the stationarity.
