Electronic Communications in Probability

Electronic Communications in Probability > Vol. 4 (1999) > Paper 2

Identifiability of Exchangeable Sequences with Identically Distributed Partial Sums

Steven N. Evans, University of California at Berkeley
Xiaowen Zhou, University of California at Berkeley

Abstract
Consider two exchangeable sequences $(X_k)_{k in bN}$ and $(hat{X}_k)_{k in bN}$ with the property that $S_n equiv sum_{k=1}^n X_k$ and $hat{S}_n equiv sum_{k=1}^n hat{X}_k$ have the same distribution for all $n in bN$. David Aldous posed the following question. Does this imply that the two exchangeable sequences have the same joint distributions? We give an example that shows the answer to Aldous' question is, in general, in the negative. On the other hand, we show that the joint distributions of an exchangeable sequence can be recovered from the distributions of its partial sums if the sequence is a countable mixture of i.i.d. sequences that are either nonnegative or have finite moment generating functions in some common neighbourhood of zero.

Full text: PDF

Pages: 9-13

Published on: February 22, 1999

Bibliography

D. J. Aldous and I. A. Ibragimov and J. Jacod, 'Ecole d''Et'e de Probabilit'es de Saint-Flour XIII, Lect. Notes in Math., page 20. Number 1117. Springer-Verlag, 1983. Math Review link
W. Feller, On M"untz' theorem and completely monotone functions, Amer. Math. Monthly (75): 342-350, 1968. Math Review link

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Electronic Communications in Probability. ISSN: 1083-589X