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Random cyclic transformations of points

Published online by Cambridge University Press:  14 July 2016

J. G. Veitch  and
G. S. Watson
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Abstract

We consider the action of independent and identically distributed n × n circulants S1, S2, … on V = [v1, …, vn] whose columns are the positions of n points in ℝd. The positions of the n points after m transformations are the columns of W(m) = VS1 … Sm. We describe, in several ways, the shape of the configuration of the points W(m) as m →∞. When n = 3, 4 and d = 2, a special discussion in terms of Moebius transformations is given.

Keywords

CIRCULANTSHAPESTOCHASTIC GEOMETRY
Type
Part 6—Allied Stochastic Processes
Information
Journal of Applied Probability , Volume 23 , Issue A: Essays in Time Series and Allied Processes , 1986 , pp. 369 - 382
Copyright
Copyright © 1986 Applied Probability Trust 

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References

Berlekamp, E. R., Gilbert, E. N. and Sinders, F. W. (1965) A polygon problem. Amer. Math. Monthly 72, 233–241.CrossRefGoogle Scholar
Davis, P. J. (1979) Circulant Matrices. Wiley, New York.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications , Vol. 2, 2nd edn. Wiley, New York.Google Scholar
James, A. T. (1954) Normal multivariate analysis and the orthogonal group. Ann. Math. Statist. 25, 40–75.Google Scholar
Kendall, D. G. (1984) Shape manifolds, Procrustean metrics and complex projective spaces. Bull. London Math. Soc. 16, 81–121.Google Scholar
Muirhead, R. (1982) Aspects of Multivariate Statistical Theory. Wiley, New York.Google Scholar
Rosenman, M. (1932) Problem 3547. Amer. Math. Monthly 39, 239.Google Scholar
Rudin, W. (1970) Real and Complex Analysis. McGraw-Hill, New York.Google Scholar
Schoenberg, I. J. (1950) The finite Fourier series and elementary geometry. Amer. Math. Monthly 57, 390–404.Google Scholar
Watson, G. S. (1983a) Random triangles. In Proc. 2nd Internat. Workshop on Stereology and Stochastic Geometry , ed. Jensen, E. B. and Gundersen, J. G., Memoirs No. 6, Institute of Mathematics, University of Aarhus, 183–199.Google Scholar
Watson, G. S. (1983b) Limit theorems on high dimensional spheres and Stiefel manifolds. In Studies in Econometrics, Time Series and Multivariate Analysis , ed. Karlin, S., Amemiya, T., and Goodman, L., Academic Press, New York, 559–570.Google Scholar