Abstract
Consider a sequence \(\left\{ {{X_j}} \right\}_{j = 1}^n \) of i.i.d. uniform {0,1,… d −1}-válued random variables, and let M n,k be the number of palindromes of length k counted in an overlapping fashion; a palindrome is any word that is symmetric about its center. We prove that the distribution of M n,k can be well-approximated by that of a Poisson random variable. Similar approximations are obtained for various other random quantities of interest. We also obtain maximal and minimal rates of growth for the length L n of the longest palindrome; an Erdös-Rényi law is derived as a corollary: the length of the longest palindrome is, almost surely, of order loga n, where a is the square root of the alphabet size d. Analogous results on partial palindromes i.e. words in which a certain (non-zero) number of “mismatches” prevent symmetry about the center, have been presented by Revelle [R] in a sequel to this paper.
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Ghosh, D., Godbole, A.P. (1996). Palindromes in Random Letter Generation: Poisson Approximations, Rates of Growth,and Erdös-Rényi Laws. In: Heyde, C.C., Prohorov, Y.V., Pyke, R., Rachev, S.T. (eds) Athens Conference on Applied Probability and Time Series Analysis. Lecture Notes in Statistics, vol 114. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0749-8_7
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DOI: https://doi.org/10.1007/978-1-4612-0749-8_7
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