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.

翻译：对于一类被称为伯努利漂移的非线性时间序列，我们建立了长期方差的平滑周期图估计量的渐近正态性。该估计量仅利用原点附近窄带傅里叶频率，因此已被广泛用于局部惠特尔估计。现有的渐近正态性结果仅适用于线性时间序列，因此我们的研究显著拓展了平滑周期图估计量的适用范围。作为示例，我们将其应用于均值变化对长期依赖性的检验。此外，我们还通过模拟研究展示了该检验在非线性时间序列中的表现。