Skip to main content Accessibility help
×
Home

On the Weak form of Zipf's law

  • Wen-Chen Chen (a1)

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.

Copyright

Corresponding author

Postal address: Department of Statistics, Carnegie-Mellon University, Schenley Park, Pittsburgh, PA 15213, U.S.A.

References

Hide All
[1] Feller, W. (1966) An Introduction to Probability Theory and its Applications , Vol. II. Wiley, New York.
[2] Gnedenko, B. V. (1962) Theory of Probability. Chelsea, New York.
[3] Hill, B. M. (1970) Zipf's law and prior distributions for the composition of a population. J. Amer. Statist. Assoc. 65, 12201232.
[4] Hill, B. M. (1974) The rank-frequency form of Zipf's law. J. Amer. Statist. Assoc. 69, 10171026.
[5] Hill, B. M. and Woodroofe, M. (1975) Stronger forms of Zipf's law. J. Amer. Statist. Assoc. 70, 212219.
[6] Holst, L. (1977) Two conditional limit theorems with applications Ann. Statist. 7, 551557.
[7] Ijiri, Y. and Simon, H. A. (1975) Some distributions associated with Bose-Einstein statistics. Proc. Nat. Acad. Sci. USA 72, 16541657.
[8] Johnson, N. and Kotz, S. (1969) Discrete Distributions. Houghton Mifflin, Boston.
[9] Mosimann, J. E. (1962) On the compound multinomial distribution, the multivariate ß-distribution, and correlations among proportions. Biometrika 49, 6582.
[10] Mosimann, J. E. (1963) On the compound negative multinomial distribution and correlations among inversely sampled pollen counts. Biometrika 50, 4754.
[11] Pólya, G. (1931) Sur quelques points de la théorie des probabilités. Ann. Inst. H. Poincaré I, 117161.
[12] Simon, H. A. (1955) On a class of skew distributions. Biometrika 42, 425440.
[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, 257261.
[14] Steck, G. (1957) Limit theorems for conditional distributions. Univ. of California Publications in Statistics, Vol. 2.
[15] Woodroofe, M. and Hill, B. M. (1975) On Zipf's law. J. Appl. Prob. 12, 425434.
[16] Willis, J. C. (1922) Age and Area. Cambridge University Press.
[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, 2187.
[18] Zipf, G. K. (1949) Human Behaviour and the Principle of Least Effort. Addison-Wesley, Reading, MA.

Keywords

On the Weak form of Zipf's law

  • Wen-Chen Chen (a1)

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed