Palindromes in Random Letter Generation: Poisson Approximations, Rates of Growth,and Erdös-Rényi Laws

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 log_a 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.