Long Range Dependence, Unbalanced Haar Wavelet Transformation and Changes in Local Mean Level

Long range dependent (LRD), stationary time series have historically served to model real time series with apparent changes in local mean level. A natural tool to study changes in local mean level is unbalanced Haar wavelet transformation (UHT). In this work, UHT is used to study changes in local mean level in LRD models and several real and simulated time series exhibiting LRD. In particular, simulations for LRD models suggest that changes in local mean level occur at times essentially governed by a homogeneous Poisson arrival process, and only the local mean levels themselves inherit the LRD property of the original time series. These properties are compared with the analogous ones in several real and simulated time series. The results are mixed though generally in favor of LRD models. The approach based on UHT is also compared to several alternatives such as defining changes in local mean level through kernel smoothing. The interest throughout is mainly in very long time series such as those collected in the studies of data traffic over Internet.