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Central limit theorem for renewal theory for several patterns

Tanushev, M.S.; Arratia, R.

Journal of Computational Biology 4(1): 35-44

1997

We prove a joint central limit theorem for the vector of counts of nonoverlapping occurrences of m given words as competing renewals. Our underlying model is an i.i.d. sequence over a finite alphabet. The motivation involves restriction enzymes in DNA sequences. We give a simple explicit formula for the limit covariance. This is in terms of the matrix of overlap-matching polynomials, following works of Guibas and Odlyzko (1980), of Breen et al. (1985), and of Biggins and Cannings (1987). The corresponding central limit theorem for counts of overlapping occurrences, rather than competing renewals, was derived by Lundstrom (1990). The above is a special case of a general situation of competing renewals in which occurrences of each type individually form a renewal process, and the individual processes interact in such a way that occurrences of either of two given types also form a renewal process. There is a simple expression for the limit covariance in this general case, involving only the means and variances for each type.

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