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28 March 2024
 
  » arxiv » 1003.1187

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Flexible Sampling of Discrete Scale Invariant Markov Processes: Covariance and Spectrum
N . Modarresi ; S . Rezakhah ;
Date 5 Mar 2010
AbstractIn this paper we consider some flexible discrete sampling of a discrete scale invariant process ${X(t), tin{f R^+}}$ with scale $l>1$. By this method we plan to have $q$ samples at arbitrary points ${f s}_0, {f s}_1,..., {f s}_{q-1}$ in interval $[1, l)$ and proceed our sampling in the intervals $[l^n, l^{n+1})$ at points $l^n{f s}_0, l^n{f s}_1,..., l^n{f s}_{q-1}$, $nin {f Z}$. Thus we have a discrete time scale invariant (DT-SI) process and introduce an embedded DT-SI process as $W(nq+k)=X(l^n{f s}_k)$, $qin {f N}$, $k= 0,..., q-1$. We also consider $V(n)=ig(V^0(n),..., V^{q-1}(n)ig)$ where $V^k(n)=W(nq+k)$, as an embedded $q$-dimensional discrete time self-similar (DT-SS) process. By introducing quasi Lamperti transformation, we find spectral representation of such process and its spectral density matrix is given. Finally by imposing wide sense Markov property for $W(cdot)$ and $V(cdot)$, we show that the spectral density matrix of $V(cdot)$ and spectral density function of $W(cdot)$ can be characterized by ${R_j(1), R_j(0), j=0,..., q-1}$ where $R_j(k)=E[W(j+k)W(j)]$.
Source arXiv, 1003.1187
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