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Journal Article

The Stochastic Convergence of a Function of Sample Successive Differences

Lionel Weiss
The Annals of Mathematical Statistics
Vol. 26, No. 3 (Sep., 1955), pp. 532-536
Stable URL: http://www.jstor.org/stable/2236483
Page Count: 5

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The Stochastic Convergence of a Function of Sample Successive Differences
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Abstract

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 - ∫B A fxe-tf(x) dx=] 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.

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