###
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}

^{1} *Department of Mathematics, Yamanashi University, Takeda, Kofu 400, Japan*

^{2} *Matsushita Electric Industrial CO. LTD., 1006 Oaza-Kadoma, Kadoma, Osaka 571, Japan*

^{3} *Department 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* \in *L*^{2} was studied by a method which makes use of the convergence of
point processes. Some one-dimensional examples with *f* \notin *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.

**Source**
( TeX ,
DVI ,
PS )