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abstract:
\name{Saeid Rezakhah}
\poster{Characterization of Discrete Time Scale Invariant Markov Process}
Scale invariant processes have recently drawn attention of many researchers. Considering discrete samples requires new tools, which we have provided as 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 $\{R_{j}^H(1),R_{j}^H(0), j=0, 1, \ldots, T-1\}$, 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.
\textbf{References:}
\begin{enumerate}
\item Rozanov Y A 1967 Stationary Random Processes (San Francisco: Holden Day)
\item Borgnat P, Amblard P O and Flandrin P 2005 Scale invariances and Lamperti transformations for stochastic
processes J. Phys. A: Math. Gen. 38 2081�101
\item Modarresi N and Rezakhah S 2009 Discrete time scale invariant Markov processes arXiv.org/pdf/0905.3959v3
\item Nematollahi A R and Soltani A R 2000 Discrete time periodically correlated Markov processes Probab. Math.
Stat. 20 127�40
\end{enumerate}
\hrulefill\newline
\address{ Faculty of
Mathematics and Computer Science, Amirkabir University of
Technology, 424 Hafez Avenue, Tehran 15914, Iran }
\email{rezakhah@aut.ac.ir} |
