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What is the distribution of the distance between a specific word in a Text which is generated by a markov process?

For example for a text which is generated by a multinomial distribution over words, this distribution of distance will be a Geometric distribution. Thanks.

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With infinitely many states, any distribution is possible, as demonstrated by a Markov chains on $N$ which jump from $n$ only to $n+1$ or $0$.

If the Markov chain is has finitely many states, or is reversible, many distributions are possible, and there is no complete characterization of which distributions can or cannot appear. In the former case you must have exponential tail. In the latter, there are also some results in certain cases, see e.g. this paper: http://arxiv.org/abs/1009.1438.

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