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

TL;DR

This paper introduces a novel approach to analyze discrete scale invariant processes by examining their covariance structures and spectral densities, especially under Markov assumptions, to better understand their multi-dimensional self-similar properties.

Contribution

It develops a new spectral analysis framework for discrete scale invariant processes, linking covariance functions to spectral density matrices under Markov assumptions.

Findings

Spectral density matrix characterized by covariance functions

Embedded process analysis reveals scale-invariant properties

Markov property simplifies spectral density characterization

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