ZIPFS LAW, THE CENTRAL-LIMIT-THEOREM, AND THE RANDOM DIVISION OF THE UNIT INTERVAL

It is shown that a version of Mandelbrot's monkey-at-the-typewriter mo del of Zipf's inverse power law is directly related to two classical a reas in probability theory: the central limit theorem and the ''broken stick'' problem, i.e., the random division of the unit interval. the connection to the central limit theorem is proved using a theorem on r andomly indexed sums of random variables [A. Gut, Stopped Random walks : Limit Theorems and Applications (Springer, New York, 1987)]. This re veals an underlying log-normal structure of pseudoword probabilities w ith an inverse power upper tail that clarifies a point of confusion in Mandelbrot's work. An explicit asymptotic formula for the slope of th e log-linear rank-size law in the upper tail of this distribution is a lso obtained. This formula relates to known asymptomatic results conce rning the random division of the unit interval that imply a slope valu e approaching -1 under quite general conditions. The role of size-bias ed sampling in obscuring the bottom part of the distribution is explai ned and connections to related work are noted.