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| Paper IPM / M / 11318 |
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| Abstract: | |
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In this paper we consider a discrete scale invariant (DSI) process {X (t), t ∈ 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 ∈ 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 ∈ 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 ∈ 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), RHj(0),j=0,1,...,T−1}, where RHj(τ) is the covariance function of jth and (j + τ)th observations of the process.
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