On parallelism in the ensemble sense between time-series models and discrete wavelet transforms of stochastic signals

The work is concerned with wavelet transforms (WT) of colored (correlated) discrete stochastic signals (time-series) and their relation to AR/ARMA models of the same signals. It derives the relations between AR/ARMA models of WT coefficients and AR/ARMA model of the signal, which eliminates the need to actually perform the WT of such signals in order to derive models of WT coefficients. The work explains how to arrive at the WT coefficient ARMA models from the signal's ARMA model and vice-versa to show that WT properties of the ensemble are fully predictable from the signal's AR/ARMA model. In particular, the authors have shown that from AR/ARMA parameters of the stochastic signal alone, one can derive a realization of the WT coefficients of that stochastic signal and that by invoking the inverse WT on those coefficients, one then retrieves a stochastic signal whose AR/ARMA structure is the same as that of the original signal. It is noted that for a stochastic signal, signal parameters, rather than a particular realization, convey the information on the signal.