Stationarization of stochastic sequences with wide-sense stationary increments or jumps by discrete wavelet transforms

Owing to most physical phenomena observed as nonstationary processes and the form of discrete sequences, it becomes realistic to process the nonstationary sequences in the laboratory if there exists a bi jective transformation for stationarization. In this work, our study is emphasized on the class of nonstationary one-dimensional random sequences with wide-sense stationary increments (WSSI), wide-sense stationary jumps (WSSJ) and a famous case, the fractional Brownian motion (FBM) process. Also, the concept of linear algebra is applied to process the stationarization concisely. Our goal is to derive a stationarization theorem developed by linear operators such that a nonstationary sequence with WSSI/WSSJ may be stationarized by an easily realizable perfect reconstruction-quadrature mirror filter structure of the discrete wavelet transform. Some examples for FBM processes and nonstationary signals generated by autoregressive integrated moving average models are provided to demonstrate the stationarization.