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Uniform order statistics property and ℓ-spherical densities

Published online by Cambridge University Press:  18 December 2007

Moshe Shaked
Affiliation:
Department of MathematicsUniversity of ArizonaTucson, AZ 85721 E-mail: shaked@math.arizona.edu
Fabio Spizzichino
Affiliation:
Department of MathematicsUniversity “La Sapienza”Piazzale Aldo Moro, 5 00815 Rome, Italy
Florentina Suter
Affiliation:
Department of MathematicsUniversity of BucharestAcademiei 14, 010014 Bucharest, Romania
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Abstract

Type
Errata
Copyright
Copyright © Cambridge University Press 2008

(This article appeared in Volume 18, Number 3, 2004 pages 275–297)

Professor Taizhong Hu has pointed out to us that the interpretation that follows the definition of the UOSP(≤) property in page 287 in the above paper is incorrect. As a consequence, Theorem 4.7 and Remark 4.8 in the above paper need to be modified.

In order to do that, we replace the definition of the UOSP(≤) property in the above paper by two definitions that are given below. Let the discrete random variables X 1, X 2, … be such that P{0 ≤ X 1 ≤ X 2 ≤ ···} = 1. We say that these X i's have the UOSP1(≤) property if for discrete 0 ≤ x 1 ≤ x 2 ≤ ··· ≤ x k ≤ t we have

where, for l ∈ {0, 1, … , t}, j l is the number of values in {x 1, x 2, … , x k} that are equal to l. That is, conditional on X k ≤ t and X k+1 > t, the random variables X 1 ≤ X 2 ≤ ··· ≤ X k are distributed as order statistics of a sample of size k drawn from the set {0, 1, … , t} with replacement. On the other hand, we say that these X i's have the UOSP2(≤) property if for discrete 0 ≤ x 1 ≤ x 2 ≤ ··· ≤ x k ≤ t we have

this is the definition of the UOSP(≤) property given in the original paper. The meaning of this definition is that conditional on X k ≤ t and X k+1 > t, the random variables X 1 ≤ X 2 ≤ ··· ≤ X k are distributed as order statistics of a sample of size k drawn from the set {0, 1, … , t} with double replacement; see (23) in page 184 of de Finetti (Reference de Finetti1975) and see also Exercise 1.62 in page 41 of Spizzichino (Reference Spizzichino2001).

With these definitions we first note that Proposition 4.2 in the original paper remains correct if UOSP(≤) is understood to mean UOSP2(≤).

Next, let {B(t), t = 0, 1, … } be a nondecreasing discrete-time discrete-state random process as described in page 291 of the original paper, and let T 1, T 2, … be the corresponding “unit jump” times, again, as described in page 291 of the original paper. We say that the process {B(t), t = 0, 1, …} has the UOSP1(≤) [respectively, UOSP2(≤)] property if T 1, T 2, … have the UOSP1(≤) [respectively, UOSP2(≤)] property. Then

  • item (i) in Theorem 4.7 of the original paper, with UOSP1(≤) instead of UOSP(≤), is equivalent to item (ii) of that theorem, and

  • item (i) in Theorem 4.7 of the original paper, with UOSP2(≤) instead of UOSP(≤), is equivalent to each of the items (iii), (iv), and (v) of that theorem.

Finally there are a couple of minor corrections in Remark 4.8. In lines 1 and 8 on page 295, UOSP2(≤) should replace UOSP(≤). The claim that the processes from Theorem 4.5 satisfy “a version of the statement in Theorem 4.7(ii)” is not true.

References

1.de Finetti, B. (1975). Theory of Probability, Volume 2. John Wiley & Sons, London.Google Scholar
2.Spizzichino, F. (2001). Subjective Probability Models for Lifetimes. Chapman & Hall/CRC, Boca Raton.CrossRefGoogle Scholar
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