IJPAM: Volume 13, No. 3 (2004)

A CONTRIBUTION TO RESULTS ON RANDOM
PARTITIONS OF THE SEGMENT

Milena Bieniek$^1$, Dominik Szynal$^2$
$^1$Department of Statistics and Economics
Faculty of Economics
Maria Curie-Sk\lodowska University
Pl. Marii Curie-Sk\lodowskiej 5, 20-031, Lublin, POLAND
e-mail: milena@ramzes.umcs.lublin.pl
$^2$Institute of Mathematics
Maria Curie-Sk\lodowska University
Pl. Marii Curie-Sk\lodowskiej 1, 20-031, Lublin, POLAND
e-mail: szynal@golem.umcs.lublin.pl


Abstract.Let $X_1,\dotsc,X_{k-1}$ be a sequence of independent random variables uniformly distributed on the interval $[0,1]$ and let $\Delta_k$ and $\delta_k$ denote the lengths of the greatest and the smallest interval of a partition of $[0,1]$ by the points $X_1,\dotsc,X_{k-1}$. In the paper we give the moments and the central moments for $\Delta_k$, $\delta_k$, $\delta_k/\Delta_k$ and $\Delta_k-\delta_k$. The moments for $D_k=\max\{X_1,\dotsc,X_k\}$ and for the quasi-range $W_{k,r}$ from the exponential distribution are also discussed. Moreover, we investigate the asymptotic behaviour of $(D_t^*/\log t)$, where $D_t^*$ is the diameter of a partition of the interval $[0,t]$ by renewal moments of a standard Poisson process.

Received: March 18, 2004

AMS Subject Classification: 62G30, 60K05, 60E99, 60G99, 60F15, 60F25

Key Words and Phrases: ordered statistics, spacings, range, diameter, Poisson process, renewal moments, hazard function, mean residual life function, Polygamma function, convergence in probability, in mean and almost sure

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2004
Volume: 13
Issue: 3