Mathematics – Probability
Scientific paper
2009-05-25
Mathematics
Probability
20 pages
Scientific paper
Scale invariant processes have recently drawn attention of many researchers. In this paper we study discrete time self-similar and scale invariance process by introducing some special quasi Lamperti transform. We study a discrete scale invariant Markov process $\{X(t), t\in {\bf R^+}\}$ with scale $l>1$ and consider to have some fix number of observations in every scale, say $T$, and to get our samples at discrete points $\alpha^k,\; k\in {\bf Z}$, where $\alpha$ is obtained by the equality $l=\alpha^T$. So we provide a discrete time scale invariant Markov (DT-SIM) process $X(\cdot)$ with parameter space $\{\alpha^k, k\in {\bf Z}\}$. We show that the covariance structure of DT-SIM process is characterized by the values of $\{R_{j}^H(1),R_{j}^H(0), j=0, 1, \..., T-1\}$, where $R_j^H(k)$ is the covariance function of $j$th and $(j+k)$th observation of the process. We introduce discrete scale invariant autoregressive 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 $\alpha^T$, we introduce $T$-dimensional self-similar Markov process, and caracterize its spectral density matrix.
Modarresi N. .
Rezakhah S. .
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