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Moments of particle size distributions under sequential breakage with applications to species abundance

Published online by Cambridge University Press:  14 July 2016

Andrew F. Siegel  and
George Sugihara
Andrew F. Siegel*
Affiliation:
Princeton University
George Sugihara*
Affiliation:
Princeton University
*
∗ Postal address: Department of Statistics, Princeton University, Princeton, NJ 08544, U.S.A.
∗∗ Postal address: Department of Biology, Princeton University, Princeton, NJ 08544, U.S.A.
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Abstract

The sequential broken stick model has appeared in numerous contexts, including biology, physics, engineering and geology. Kolmogorov showed that under appropriate conditions, sequential breakage processes often yield a lognormal distribution of particle sizes. Of particular interest to ecologists is the observed variance of the logarithms of the sizes, which characterizes the evenness of an assemblage of species. We derive the first two moments for the logarithms of the sizes in terms of the underlying distribution used to determine the successive breakages. In particular, for a process yielding n pieces, the expected sample variance behaves asymptotically as log(n). These results also yield a new identity for moments of path lengths in random binary trees.

Keywords

BROKEN STICK MODELLOGNORMAL DISTRIBUTIONECOLOGYNICHE HIERARCHY MODELRANDOM BINARY TREES
Type
Short Communications
Information
Journal of Applied Probability , Volume 20 , Issue 1 , March 1983 , pp. 158 - 164
Copyright
Copyright © Applied Probability Trust 1983 

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References

Aitchison, J. and Brown, J. A. C. (1968) The Lognormal Distribution, with Special Reference to its Uses in Economics, 2nd edn. Cambridge University Press, London.Google Scholar
Brown, G. and Sanders, J. W. (1981) Lognormal genesis. J. Appl. Prob. 18, 542–547.Google Scholar
Bulmer, M. G. (1974) On fitting the Poisson lognormal distribution to species-abundance data. Biometrics 30, 101–110.Google Scholar
Camp, B. H. (1938) Notes on the distribution of the geometric mean. Ann. Math. Statist. 9, 221–226.Google Scholar
Clark, P. J. (1964) On the number of individuals per occupation in a human society. Ecology 45, 367–372.Google Scholar
Cramér, H. (1946) Mathematical Methods of Statistics. Princeton Mathematical Series 9, Princeton University Press.Google Scholar
Davies, G. R. (1946) Pricing and price levels. Econometrica 14, 219–226.Google Scholar
Epstein, B. (1947) The mathematical description of certain breakage mechanisms leading to the logarithmico-normal distribution. J. Franklin Inst. 224, 471–477.CrossRefGoogle Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. Wiley, New York.Google Scholar
Gibrat, R. (1931) Les inegalités économiques. Librairie du Recueil Sirey, Paris.Google Scholar
Hansen, E. R. (1975) A Table of Series and Products. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Herdan, G. (1953) Small Particle Statistics. Elsevier, Amsterdam.Google Scholar
Kapteyn, J. C. (1916) Skew frequency curves in biology and statistics. Rec. Trav. Botaniques Néerlandais 13, 105–158.Google Scholar
Knuth, D. E. (1973) The Art of Computer Programming Vols. 1, 3. Addison-Wesley, Reading, MA.Google Scholar
Kolmogorov, A. N. (1941) Über das logarithmisch normale Verteilungsgesetz der Dimensionen der Teilchen bei Zerstückelung. C.R. Acad. Sci. U.R.S.S. 31, 99–101.Google Scholar
May, R. M. (1975) Patterns of species abundance and diversity. In Ecology and Evolution of Communities, ed. Cody, M. L. and Diamond, J. M., Harvard University Press, Cambridge, MA, 81–120.Google Scholar
Patrick, R. (1968) The structure of diatom communities in similar ecological conditions. Amer. Natur. 102, 173–183.Google Scholar
Pielou, E. C. (1975) Ecological Diversity. Wiley, New York.Google Scholar
Preston, F. W. (1962) The canonical distribution of commonness and rarity: Part I. Ecology 43, 185–215.CrossRefGoogle Scholar
Sugihara, G. (1980) Minimal community structure: an explanation of species abundance patterns. Amer. Natur. 116, 770–787.Google Scholar
Yuan, P. T. (1933) On the logarithmic frequency distribution and the semilogarithmic correlation surface. Ann. Math. Statist. 4, 30–74.Google Scholar