Characterization of Discrete Time Scale Invariant Markov Sequences

TL;DR

This paper characterizes discrete scale invariant Markov sequences, explores their covariance and spectral properties, and introduces a new Hurst parameter estimation method with improved performance over existing techniques.

Contribution

It provides a theoretical framework for DSI Markov sequences, including covariance characterization, spectral density estimation, and a novel Hurst parameter estimator.

Findings

Covariance functions are characterized by variance and adjacent sample covariances.

Spectral density matrix estimation method is developed for multi-dimensional self-similar Markov processes.

The new Hurst parameter estimator outperforms maximum likelihood in simulations.

Abstract

By considering special sampling of discrete scale invariant (DSI) processes we provide a sequence which is in correspondence to multi-dimensional self-similar process. By imposing Markov property we show that the covariance functions of such discrete scale invariant Markov (DSIM) sequences are characterized by variance, and covariance of adjacent samples in the first scale interval. We also provide a theoretical method for estimating spectral density matrix of corresponding multi-dimensional self-similar Markov process. Some examples such as simple Brownian motion with drift and scale invariant autoregressive model of order one are presented and these properties are investigated. By simulating DSIM sequences we provide visualization of their behavior and investigate these results. Finally we present a new method to estimate Hurst parameter of DSI processes and show that it has much better performance than maximum likelihood method for simulated data.