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.
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