Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-05T16:55:16.352Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Weak form of Zipf's law

Published online by Cambridge University Press:  14 July 2016

Wen-Chen Chen
Wen-Chen Chen*
Affiliation:
Carnegie–Mellon University
*
∗Postal address: Department of Statistics, Carnegie-Mellon University, Schenley Park, Pittsburgh, PA 15213, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

Zipf's laws are probability distributions on the positive integers which decay algebraically. Such laws have been shown empirically to describe a large class of phenomena, including frequency of words usage, populations of cities, distributions of personal incomes, and distributions of biological genera and species, to mention only a few. In this paper we present a Dirichlet–multinomial urn model for describing the above phenomena from a stochastic point of view.

We derive the Zipf's law under certain regularity conditions; some limit theorems are also obtained for the urn model under consideration.

Keywords

ASYMPTOTIC NORMALITIESCONVERGENCE IN PROBABILITYCONVERGENCE IN DISTRIBUTIONDIRICHLET–MULTINOMIAL URN MODELMAXIMUM LIKELIHOOD ESTIMATEZIPF'S LAW
Type
Research Papers
Information
Journal of Applied Probability , Volume 17 , Issue 3 , September 1980 , pp. 611 - 622
Copyright
Copyright © Applied Probability Trust 1980 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Feller, W. (1966) An Introduction to Probability Theory and its Applications , Vol. II. Wiley, New York.Google Scholar
[2] Gnedenko, B. V. (1962) Theory of Probability. Chelsea, New York.Google Scholar
[3] Hill, B. M. (1970) Zipf's law and prior distributions for the composition of a population. J. Amer. Statist. Assoc. 65, 1220–1232.Google Scholar
[4] Hill, B. M. (1974) The rank-frequency form of Zipf's law. J. Amer. Statist. Assoc. 69, 1017–1026.CrossRefGoogle Scholar
[5] Hill, B. M. and Woodroofe, M. (1975) Stronger forms of Zipf's law. J. Amer. Statist. Assoc. 70, 212–219.Google Scholar
[6] Holst, L. (1977) Two conditional limit theorems with applications Ann. Statist. 7, 551–557.Google Scholar
[7] Ijiri, Y. and Simon, H. A. (1975) Some distributions associated with Bose-Einstein statistics. Proc. Nat. Acad. Sci. USA 72, 1654–1657.Google Scholar
[8] Johnson, N. and Kotz, S. (1969) Discrete Distributions. Houghton Mifflin, Boston.Google Scholar
[9] Mosimann, J. E. (1962) On the compound multinomial distribution, the multivariate ß-distribution, and correlations among proportions. Biometrika 49, 65–82.Google Scholar
[10] Mosimann, J. E. (1963) On the compound negative multinomial distribution and correlations among inversely sampled pollen counts. Biometrika 50, 47–54.Google Scholar
[11] Pólya, G. (1931) Sur quelques points de la théorie des probabilités. Ann. Inst. H. Poincaré I, 117–161.Google Scholar
[12] Simon, H. A. (1955) On a class of skew distributions. Biometrika 42, 425–440.Google Scholar
[13] Skellam, J. G. (1948) A probability distribution derived from the binomial distribution by regarding the probability of success as variable between the sets of trials. J. R. Statist. Soc. 10, 257–261.Google Scholar
[14] Steck, G. (1957) Limit theorems for conditional distributions. Univ. of California Publications in Statistics, Vol. 2.Google Scholar
[15] Woodroofe, M. and Hill, B. M. (1975) On Zipf's law. J. Appl. Prob. 12, 425–434.Google Scholar
[16] Willis, J. C. (1922) Age and Area. Cambridge University Press.Google Scholar
[17] Yule, G. U. (1924) A mathematical theory of evolution, based on conclusions of Dr. J. C. Willis, F.R.S. Phil. Trans. R. Soc . London. B 213, 21–87.Google Scholar
[18] Zipf, G. K. (1949) Human Behaviour and the Principle of Least Effort. Addison-Wesley, Reading, MA.Google Scholar