Statistical inference for mean and variance of oscillatory processes

When constructing confidence intervals for the mean and variance of a stationary continuous-time stochastic process, two approaches have been considered in the literature: one based on the so-called long-run variance of the process and its square, and the other based on the so-called self-normalization. These approaches are revisited here in the context of random oscillatory processes such as random (non-)linear oscillators and related models with particular attention to the problem of estimating the non-zero long-run variances of the processes. The case of the zero long-run variance, which has been studied and is quite different, is also considered. The approaches are extended to the situation where multiple independent records of the stochastic process are available, for example, by introducing an estimator of the long-run variance which improves on other natural candidate estimators. Finally, a simulation study is provided to assess the performance of the proposed methods in estimating the long-run variances and constructing confidence intervals, and a data example is considered.