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

Abstract

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 $V(n)=(V^0(n),...,V^{q-1}(n))$ 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 $R_j(k)=E[W(j+k)W(j)]$.

14 pages