B-Splines and Linear Combinations of Uniform Order Statistics

It is shown in this document that the density of the sum of n independent random variables uniformly distributed on unequal intervals is given by a linear combination of n! B-splines with constant coefficients. Another useful representation of the same density is given using de Boor's definition of the discrete B-spline. It is also shown that the B-spline is the density of a linear combination of order statistics from the uniform distribution on (0,1). This interpretation of the B-spline allows one to establish easily validity of recurrence relations for densities and moments of linear combinations of order statistics as well as to state limit theorems both for B-splines and linear combinations. Examples given illustrate how asymptotic results for B-splines may be applied to linear combinations of order statistics. Using limit theorems of probability theory two examples of Curry and Schoenberg (1966) for B-splines are derived even in somewhat greater generality.