Testing of tempered fractional Brownian motions

We propose here a testing methodology based on the autocovariance, detrended moving average, and time-averaged mean-squared displacement statistics for tempered fractional Brownian motions (TFBMs) which are related to the notions of semi-long range dependence and transient anomalous diffusion. In this framework, we consider three types of TFBMs: two with a tempering factor incorporated into their moving-average representation, and one with a tempering parameter added to the autocorrelation formula. We illustrate their dynamics with the use of quantile lines. Using the proposed methodology, we provide a comprehensive power analysis of the tests. It appears that the tests allow distinguishing between the tempered processes with different Hurst parameters.