The Stochastic Convergence of a Function of Sample Successive Differences

Let f (x) be a bounded density function over the finite interval A, B with at most a finite number of discountinities. Let X₁, X₂, , Xₙ be independent chance variables each with the density f (x). Define Y₁ Y₂ Y ₙ as the ordered values of X₁, X₂, , Xₙ, and Tᵢ as Y₈+₁ - Yᵢ. Also define Rₙ (t) as the proportion of the variates T₁, , T₍-₁ not greater than t / (n - 1). We shall denote 1 - BA fxe^-tf (x) dx= by S (t), and ₓ ₀ \|Rₙ (t) - S (t) \| by V (n). Then it is shown that as n increases, V (n) converges stochastically to zero. The relation of this result to other results is discussed.