Characterization of Discrete Time Scale Invariant Markov Processes S. Rezakhah and M. Modarresi ∗ Abstract Scale invariant processes have recently drawn attention of many researchers. Con- sidering discrete samples requires new tools, which we have provided as some spe- cial quasi Lamperti transform. We study a discrete scale invariant Markov process {X(t),t ∈ R + } with scale l> 1 and consider to have some fix number of observa- tions in every scale, say T , and to get our samples at discrete points α k ,k ∈ Z, where α is obtained by the equality l = α T . So we provide a discrete time scale invariant Markov (DT-SIM) process X(·) with parameter space {α k ,k ∈ Z}. We show that the covariance structure of DT-SIM process is characterized by the values of {R H j (1),R H j (0),j =0, 1,...,T − 1}, where R H j (k) is the covariance function of j th and (j + k)th observation of the process. We introduce discrete scale invariant au- toregressive process of order p, DSIAR(p) with time varying coefficients. We present two examples of DT-SIM process as DSIAR(1) and discrete time simple Brownian motion and justify our characterization result. In correspondence to our DT-SIM process with scale α T , we introduce T -dimensional self-similar Markov process. AMS 2000 Subject Classification: 60G18, 60J05, 60G12. Keywords: Discrete scale invariance; Wide sense Markov; Multi-dimensional self- similar processes. 1 Introduction 2.1The notion[10] of scale invariance or self-similarity is used as a fundamental property to handle and interpret many natural phenomena, like textures in geophysics, turbulence of fluids, data of network traffic and image processing, etc [1]. The idea is that a function is scale invariant if it is identical to any of its rescaled functions, up to some suitable renormalization of its amplitude. ∗ Faculty of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Av- enue, Tehran 15914, Iran. E-mail: rezakhah@aut.ac.ir (S. Rezakhah), namomath@aut.ac.ir (N. Modarresi). 1