LIMIT THEOREMS FOR THE MINIMUM INTERPOINT DISTANCE BETWEEN ANY PAIR OF I.I.D. RANDOM POINTS IN R^{d}

S. KANAGAWA^{1}, Y. MOCHIZUKI^{2} AND H. TANAKA^{3}

1Department of Mathematics, Yamanashi University, Takeda, Kofu 400, Japan 2Matsushita Electric Industrial CO. LTD., 1006 Oaza-Kadoma, Kadoma, Osaka 571, Japan

3Department of Mathematics, Faculty of Science and Technology, Keio University, Kohoku-ku, Yokohama, Kanagawa 223, Japan

(Received May 1, 1989; revised November 5, 1990)

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.

Key words and phrases: Minimum interpoint distance, Poisson point process, compensator, Skorohod J_{1}-topology.