  For a broad class of nonlinear time series known as Bernoulli shifts, we
establish the asymptotic normality of the smoothed periodogram estimator of the
long-run variance. This estimator uses only a narrow band of Fourier
frequencies around the origin and so has been extensively used in local Whittle
estimation. Existing asymptotic normality results apply only to linear time
series, so our work substantially extends the scope of the applicability of the
smoothed periodogram estimator. As an illustration, we apply it to a test of
changes in mean against long-range dependence. A simulation study is also
conducted to illustrate the performance of the test for nonlinear time series.
