A class of birth processes arising from a work of B. Mandelbrot is investigated. It involves a stationary Markov chain in which the states are words made up of letters from a certain prescribed alphabet. In the transition from the word Xn, the n-th term of the chain. to the word Xn+1, each letter of the former is replaced by a random subword. The subwords replacing neighboring letters may be statistically dependent. Let | Xn | be the length of Xn. Under suitable hypotheses there exists a > l such that, with probability 1.
-n | Xn | has a finite. nonzero limit. In addition, given any word w, and denoting by
n(w) the number of times that w appears as a subword of Xn, the ratio
n(w) / | Xn | tends to a constant with probability 1. Among other applications. Mandelbrot uses these birth processes to construct a variety of fractal random curves, and he makes conjectures concerning their fractal Hausdorff dimension. Proofs of generalized forms of some of these conjectures are given.
By: Jacques Peyriere
Published in: RC7417 in 1978
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