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. 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 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.
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