## The Annals of Mathematical Statistics

### The Stochastic Convergence of a Function of Sample Successive Differences

Lionel Weiss

#### 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.

#### Article information

Source
Ann. Math. Statist., Volume 26, Number 3 (1955), 532-536.

Dates
First available in Project Euclid: 28 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177728501

Digital Object Identifier
doi:10.1214/aoms/1177728501

Mathematical Reviews number (MathSciNet)
MR70874

Zentralblatt MATH identifier
0068.12103

JSTOR
links.jstor.org

#### Citation

Weiss, Lionel. The Stochastic Convergence of a Function of Sample Successive Differences. Ann. Math. Statist. 26 (1955), no. 3, 532--536. doi:10.1214/aoms/1177728501. https://projecteuclid.org/euclid.aoms/1177728501