Spectral analysis of multi-dimensional self-similar Markov processes

In this paper we consider a discrete scale invariant (DSI) process {X(t), t in R+} with scale l > 1. We consider a fixed number of observations in every scale, say T, and acquire our samples at discrete points αk, k in W, where α is obtained by the equality l = αT and W = {0, 1, ...}. We thus provide a discrete time scale invariant (DT-SI) process X(.) with the parameter space {αk, k in W}. We find the spectral representation of the covariance function of such a DT-SI process. By providing the harmonic-like representation of multi-dimensional self-similar processes, spectral density functions of them are presented. We assume that the process {X(t), t in R+} is also Markov in the wide sense and provide a discrete time scale invariant Markov (DT-SIM) process with the above scheme of sampling. We present an example of the DT-SIM process, simple Brownian motion, by the above sampling scheme and verify our results. Finally, we find the spectral density matrix of such a DT-SIM process and show that its associated T-dimensional self-similar Markov process is fully specified by {RHj(1), RjH(0), j = 0, 1, ..., T - 1}, where RHj(τ) is the covariance function of jth and (j + τ)th observations of the process.