  We study statistical inferences for a class of modulated stationary processes
with time-dependent variances. Due to non-stationarity and the large number of
unknown parameters, existing methods for stationary, or locally stationary,
time series are not applicable. Based on a self-normalization technique, we
address several inference problems, including a self-normalized central limit
theorem, a self-normalized cumulative sum test for the change-point problem, a
long-run variance estimation through blockwise self-normalization, and a
self-normalization-based wild bootstrap. Monte Carlo simulation studies show
that the proposed self-normalization-based methods outperform
stationarity-based alternatives. We demonstrate the proposed methodology using
two real data sets: annual mean precipitation rates in Seoul from 1771-2000,
and quarterly U.S. Gross National Product growth rates from 1947-2002.
