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Article overview
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Flexible Sampling of Discrete Scale Invariant Markov Processes: Covariance and Spectrum | N . Modarresi
; S . Rezakhah
; | Date: |
5 Mar 2010 | Abstract: | In 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 | Services: | Forum | Review | PDF | Favorites |
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