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Paper IPM / M / 11318  


Abstract:  
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 α^{k}, k ∈ W where α is obtained by the equality l=α^{T} and W={0,1,...}. We thus provide a discrete time scale invariant (DTSI) process X(.) with the parameter space α^{k}, k ∈ W. We find the spectral representation of the covariance function of such a DTSI process. By providing the harmoniclike representation of multidimensional selfsimilar 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 (DTSIM) process with the above scheme of sampling. We present an example of the DTSIM process, simple Brownian motion, by the above sampling scheme and verify our results. Finally, we find the spectral density matrix of such a DTSIM process and show that its associated Tdimensional selfsimilar Markov process is fully specified by {R^{H}_{j}(1), R^{H}_{j}(0),j=0,1,...,T−1}, where R^{H}_{j}(τ) is the covariance function of jth and (j + τ)th observations of the process.
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