A new structure for analyzing discrete scale invariant processes: Covariance and Spectra

Improving the efficiency of discrete time scale invariant (DSI) processes, we consider some flexible sampling of a continuous time DSI process ${X(t), t\in{R^+}}$ with scale $l>1$, which is in correspondence to some multi-dimensional self-similar process. So we consider $q$ samples at arbitrary points $s_0, s_1, ..., s_{q-1}$ in interval $[1, l)$ and proceed in the intervals $[l^n, l^{n+1})$ at points $l^n s_0,l^n s_1, ..., l^n s_{q-1}$, $n\in Z$. So we study an embedded DT-SI process $W(nq+k)=X(l^n s_k)$, $q\in N$, $k= 0, ..., q-1$, and its multi-dimensional self-similar counter part $V(n)=\big(V^0(n), ..., V^{q-1}(n)\big)$ where $V^k(n)=W(nq+k)$. We study spectral representation of such process and obtain its spectral density matrix. Finally by imposing wide sense Markov property on $W(\cdot)$ and $V(\cdot)$, we show that the spectral density matrix of $V(\cdot)$ can be characterized by ${R_j(1), R_j(0), j=0, ..., q-1}$ where $R_j(k)=E[W(j+k)W(j)]$.