Summary
If \(\forall n\sum\limits_\pi {P(X_{\pi _1 } < ... < } X_{\pi _n } ) = 1\) and \(\forall \pi ,n,{\text{ }}P(X_{\pi _1 } < ... < X_{\pi _n } ) = P(Y_{\pi _1 } < ... < Y_{\pi _n } )\) then P(n −1·[δ(Y 1)+⋯+δ(Y n )] converges to cnts. law on R 1) = P(n −1·[δ(Y 1)+⋯+δ(Y n )] converges to a cnts. law on R 1). Thus if \(P(X_{\pi _1 } < ... < X_{\pi _n } ) = (n!)^{ - 1} \forall \pi ,n\),n then n −1[δ(X 1)+...+δ(X n )] converges a.s. The main result here generalizes this: Let X n(1) , X n(2) ,..., X n(n) be the order statistics associated with X 1, X 2,⋯,X n. Define random variables Z 1,Z 2,⋯ by {Z n =i}={X n =X n(i) }. Then if Z 1,Z 2,Z 3, ⋯ are independent and P(Zn≦i)≦i/n, and {X i} is bounded, n −1·[δ(X 1)+⋯+δ(X n)] converges a.s.
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Gutmann, S. Interval-dividing processes. Z. Wahrscheinlichkeitstheorie verw Gebiete 57, 339–347 (1981). https://doi.org/10.1007/BF00534828
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DOI: https://doi.org/10.1007/BF00534828