Characterization of Discrete Time Scal Invariant Markov Processes

Characterization of Discrete Time Scal Invariant Markov Processes

Scale invariant processes have recently drawn attention of many researchers. We study a discrete scale invariant (DSI) process { X(t), t e R. with scale t > 1 and consider to have some fix number of observations in every scale, say T, and to get our samples at discrete points of, ke Z. where () is obtained by the equality l = oʹ. So we provide a basi s discrete scale invariant Markov (DSIM) sequence X () with parameter space {cf. k e Z}. We show that the covariance structure of DSIM sequence is characterized by the values of {R(1), R(0), j = 0, 1,..., T -1}, where R(k) is the covariance function of ith and ( i + k)th observation of the proc In correspondence to the DSIM sequence with scale cyʹ, we introduce T-dimensional self-similar Markov process. By introducing some multi-dimensional selfsimilar process corresponding to the DSIM sequence, we present spectral density matrix of stech processes. Some Examples of Stech processes like simple Brownian motion and scale invariant autoregressive, AR(1) model are presented and these properties are investigated. By simulating DSIM process we provide visualization of their behavior and investigate results of the paper. Finally we present new method to estimate Hurst parameters of DSI and selfsimilar process and apply it to the simulated data.