Tree Indexed Markov Processes and Long Range Dependency

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 [I]. 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 selfsimilarity property. These models are motivated in part by recent evidence that suggests some communications network traffic may exhibit such behaviour. 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.