This paper gives an elementary proof that, under some general assumptions, the number of parts a convex set in Rd is divided into by a set of independent identically distributed hyperplanes is asymptotically normally distributed. An example is given where the distribution of hyperplanes is ‘too singular' to satisfy the assumptions, and where a different limiting distribution appears.

Let X1, X2, ···, Xn be a random sample of size n from a population with probability density function f(x), x > 0, and let X1, n < X2, n < ··· < Xn, n be the associated order statistics. A characterization of the exponential distribution is shown by considering identical distribution of the random variables (n − i + 1)(Xi, n − Xi−1, n) and (n − i)(Xi+1, n − Xi, n) for one i and one n with 2 ≦ i ˂ n.

The steady-state distribution of the inventory position for a continuous review (s, S) inventory system is derived. The demand for items in inventory is dependent on an external environment. During an interval of time in which the environment is in a fixed state, the demand follows a discrete-valued compound Poisson process. The parameters of the compound Poisson process depend completely on the state of the environment. The environmental process is modeled as a continuous-time Markov process.