Abstract
We investigate the behavior of the periods and border lengths of random words over a fixed alphabet. We show that the asymptotic probability that a random word has a given maximal border length k is a constant, depending only on k and the alphabet size ℓ.
We give a recurrence that allows us to determine these constants with any required precision. This also allows us to evaluate the expected period of a random word.
For the binary case, the expected period is asymptotically about nMINUS SIGN 1.641. We also give explicit formulas for the probability that a random word is unbordered or has maximum border length one.