Tree indexed Markov processes and long range dependency
L.B. White · AIP conference proceedings · 2000
This paper describes the second order statistics of a finite state Markov process indexed on a binary tree. Such models are the discrete state analogues of the continuous state Gauss-Markov processes as described by Basseville et al. [1]. Such processes are termed tree-indexed processes. The idea is to use the leaf nodes of the tree at a specified depth, as indices for a time series, and to derive a probabilistic model for this time series. The paper shows that such processes possess covariance functions which decay as a power law thus exhibiting a long range dependent (LRD) or self-similarity property. These models are motivated in part by recent evidence that suggests some communications network traffic may exhibit such behavior. However, the processes are highly non-stationary in nature. The paper poses as an open question whether there exists a modification of the tree structure which permits the leaf node process to be stationary but retains the LRD property.