Abstract
The limit theorem for the minimum interpoint distance between any pair of i.i.d. random points in R d with common density f∈L 2 was studied by a method which makes use of the convergence of point processes. Some one-dimensional examples with f∉L 2 (including the cases Beta and Gamma distributions) were also considered.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aly, A. A., Beirlant, J. and Horváth, L. (1984). Strong and weak approximations of k-spacings processes. Z. Wahrsch. Verw. Gebiete, 66, 461–484.
Deheuvels, P., Einmahl, J. H. J. and Mason, D. M. (1988). The almost sure behavior of maximal and minimal multivariate k n-spacings, J. Multivariate Anal., 24, 155–176.
Durrett, R. and Resnick, S. I. (1978). Functional limit theorems for dependent variables, Ann. Probab., 6, 829–846.
Jagers, P. (1974). Aspects of random measures and point processes, Advances in Probability and Related Topics, Vol. 3, 179–239, Dekker, New York.
Kasahara, Y. and Watanabe, S. (1986). Limit theorems for point processes and their functionals, J. Math. Soc. Japan, 38, 543–574.
Lévy, P. (1939). Sur la division d'un segment par des points choisis au hasard, C. R. Acad. Sci. Paris, 208, 147–149.
Lindvall, T. (1973). Weak convergence of probability measures and random functions in the function space D[0, ∞], J. Appl. Probab., 10, 109–121.
Molchanov, S. A. and Reznikova, A. Ya. (1982). Limit theorems for random partitions, Theory Probab. Appl., 27, 310–323.
Onoyama, T., Sibuya, M. and Tanaka, H. (1984). Limit distribution of the minimum distance between independent and identically distributed d-dimensional random variables, Statistical Extremes and Applications (ed. J.Tiago de Oliveira), 549–562, Reidel, Dordrecht, Holland.
Pyke, R. (1965). Spacings, J. Roy. Statist. Soc. Ser. B, 27, 395–446.
Rao Jammalamadaka, S. and Janson, S. (1986). Limit theorems for a triangular scheme of U-statistics with applications to inter-point distances, Ann. Probab., 14, 1347–1358.
Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes, Springer, New York.
Serfozo, R. (1982). Functional limit theorems for extreme values of arrays of independent random variables, Ann. Probab., 10, 172–177.
Silverman, B. and Brown, T. (1978). Short distances, flat triangles and Poisson limits, J. Appl. Probab., 15, 815–825.
Author information
Authors and Affiliations
About this article
Cite this article
Kanagawa, S., Mochizuki, Y. & Tanaka, H. Limit theorems for the minimum interpoint distance between any pair of i.i.d. random points in R d . Ann Inst Stat Math 44, 121–131 (1992). https://doi.org/10.1007/BF00048674
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00048674