Exact formula for the expectation of the ratio of the sum of squares by the square of the sum

Let (X-n)(n greater than or equal to 1) be a sequence of identically distributed independent nonnegative random variables satisfying P { X-1 = 0} < 1. The asymptotic behaviour of the ratio R-n = E[(X-1(2) + ... + X-n(2))/(X-1 + ... + X-n)(2)] has been studied by McLeish and O'Brien. We derive an exact representation for R-n by means of the Laplace transform phi of X-1, i.e., phi (s) = E [e(-sX1)] = integral(0)(+infinity) e(-sx) dF (x), s > 0, where F denotes the distribution function of X-1. This representation allows us to provide new proofs of several asymptotic theorems on R-n, and to extend some results obtained by those authors.