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 Yᵢ₊₁ - Yᵢ. Also define Rₙ(t) as the proportion of the variates T₁, ⋯, Tₙ₋₁ not greater than $t / (n - 1).$ We shall denote 1 - ∫BA fxe⁻ᵗᶠ⁽ˣ⁾ dx= by $S(t),$ and t 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.