The Stochastic Convergence of a Function of Sample Successive Differences

Abstract

Let $f(x)$ be a bounded density function over the finite interval [A, B] with at most a finite number of discountinities. Let $X_1, X_2, \cdots, X_n$ be independent chance variables each with the density $f(x).$ Define $Y_1 \leqq Y_2 \leqq \cdots \leqq Y _n$ as the ordered values of $X_1, X_2, \cdots, X_n,$ and $T_i$ as $Y_{i+1} - Y_i.$ Also define $R_n(t)$ as the proportion of the variates $T_1, \cdots, T_{n-1}$ not greater than $t / (n - 1).$ We shall denote $\lbrack 1 - \int^B_A fxe^{-tf(x)} dx=\rbrack$ by $S(t),$ and $\sup_{t\geqq 0} \|R_n(t) - S(t)\|$ 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.