Spectral Analysis of Multi-dimensional Self-similar Markov Processes

Abstract

In this paper we consider a discrete scale invariant (DSI) process $\{X(t),t\in {\mathbf{R}}^{+}\}$ 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 ${\alpha }^k,k\in \mathbf{W}$ where $\alpha$ is obtained by the equality $l={\alpha }^T$ and $\mathbf{W}=\{0,1,...\}$. So we provide a discrete time scale invariant (DT-SI) process $X(\cdot )$ with parameter space $\{{\alpha }^k,k\in \mathbf{W}\}$. We find the spectral representation of the covariance function of such DT-SI process. By providing harmonic like representation of multi-dimensional self-similar processes, spectral density function of them are presented. We assume that the process $\{X(t),t\in {\mathbf{R}}^{+}\}$ is also Markov in the wide sense and provide a discrete time scale invariant Markov (DT-SIM) process with the above scheme of sampling. We present an example of DT-SIM process, simple Brownian motion, by the above sampling scheme and verify our results. Finally we find the spectral density matrix of such DT-SIM process and show that its associated $T$-dimensional self-similar Markov process is fully specified by $\{R_j^H(1),R_j^H(0),j=0,1,...,T-1\}$ where $R_j^H(\tau )$ is the covariance function of $j$th and $(j+\tau )$th observations of the process.

16 pages

References in corpus (1)

• Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes

Cited by in corpus (1)

• Discretization of Continuous Time Discrete Scale Invariant Processes: Estimation and Spectra