Flexible Sampling of Discrete Scale Invariant Markov Processes: Covariance and Spectrum

Details Flexible Sampling of Discrete Scale Invariant Markov Processes: Covariance and Spectrum Flexible Sampling of Discrete Scale Invariant Markov Processes: Covariance and Spectrum

2010-03-05

arxiv.org/abs/1003.1187v1

Mathematics

Probability

14 pages

Scientific paper

In this paper we consider some flexible discrete sampling of a discrete scale invariant process $\{X(t), t\in{\bf R^+}\}$ with scale $l>1$. By this method we plan to have $q$ samples at arbitrary points ${\bf s}_0, {\bf s}_1,..., {\bf s}_{q-1}$ in interval $[1, l)$ and proceed our sampling in the intervals $[l^n, l^{n+1})$ at points $l^n{\bf s}_0, l^n{\bf s}_1,..., l^n{\bf s}_{q-1}$, $n\in {\bf Z}$. Thus we have a discrete time scale invariant (DT-SI) process and introduce an embedded DT-SI process as $W(nq+k)=X(l^n{\bf s}_k)$, $q\in {\bf N}$, $k= 0,..., q-1$. We also consider $V(n)=\big(V^0(n),..., V^{q-1}(n)\big)$ 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 $R_j(k)=E[W(j+k)W(j)]$.

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