Discrete Time Scale Invariant Markov Processes

In this paper we consider a discrete scale invariant Markov process {X(t), t ∈ R} with scale l > 1. We consider to have some fix number of observations in every scale, say T , and to get our samples at discrete points α, k ∈ W, where α is obtained by the equality l = α and W = {0, 1, . . .}. So we provide a discrete time scale invariant Markov (DT-SIM) process X(·) with parameter space {α, k ∈ W}. We present some properties of such a DT-SIM process and we show that the covariance function is characterized by the values of {R j (1), R H j (0), j = 0, 1, . . . , T − 1}, where R H j (k) is the covariance function of jth and (j + k)th observations of the process. We also define the corresponding T -dimensional self-similar Markov process and characterize its covariance matrix. AMS 2000 Subject Classification: 60G18, 60J05, 60G12.