On the clustering of independent uniform random variables

We consider the number Kn of clusters at a distance level dn ∈ (0, 1) of n independent random variables uniformly distributed in [0, 1] , or the number Kn of connected components in the random interval graph generated by these variables and dn , and, depending upon how fast dn → 0 as n → ∞ , determine the asymptotic distribution of Kn , with rates of convergence, and of related random variables that describe the cluster sizes.

[1]  D. Aldous.  Probability Approximations via the Poisson Clumping Heuristic , 1988 .

[2]  E. Godehardt,et al.  The Application of Random Coincidence Graphs for Testing the Homogeneity of Data , 1998 .

[3]  B. M. Hill,et al.  Zipf's Law and Prior Distributions for the Composition of a Population , 1970 .

[4]  T. A. Azlarov,et al.  Refinements of Yu. V. Prokhorov's theorems on the asymptotic behavior of the binomial distribution , 1987 .

[5]  A. Barbour,et al.  Poisson Approximation , 1992 .

[6]  Lars Holst,et al.  ON THE RANDOM COVERAGE OF THE CIRCLE , 1984 .

[7]  E. Godehardt.  Graphs as Structural Models: The Application of Graphs and Multigraphs in Cluster Analysis , 1988 .

[8]  Bruce M. Hill,et al.  The Rank-Frequency Form of Zipf's Law , 1974 .

[9]  B. Harris,et al.  Probability Models and Limit Theorems for Random Interval Graphs with Applications to Cluster Analysis , 1998 .

[10]  Mathew D. Penrose,et al.  Random Geometric Graphs , 2003 .

[11]  J. Seaman.  Introduction to the theory of coverage processes , 1990 .

[12]  Winfried Stute,et al.  THE OSCILLATION BEHAVIOR OF EMPIRICAL PROCESSES , 1982 .

[13]  Hans-Hermann Bock,et al.  Probabilistic Models in Partitional Cluster Analysis , 2003 .

[14]  D. Pollard.  Convergence of stochastic processes , 1984 .

[15]  Erhard Godehardt,et al.  On the connectivity of a random interval graph , 1996 .

[16]  L. Devroye.  Laws of the Iterated Logarithm for Order Statistics of Uniform Spacings , 1981 .