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This question is motivated in part by an assumption I had to make in a write-up of this one.

There are a number of well-known sufficient conditions for a measure to be determined by its moments, probably the most powerful of which is Carleman's condition. The essential idea in all of the proofs I have seen is to use the theory of analytic functions to show that some appropriate analytic function of the measure (the Fourier or Laplace transforms usually) is uniquely identified by the moments and that we can recover the measure from the function. Clearly these proofs do not work if any of the moments are infinite, since the relevant power series will diverge. A natural (kind of) converse--which I am sure is somewhere in the literature--is whether it is possible for a measure with infinite moments to be uniquely identifiable by its sequence of moments.

(This is purely heuristic) Morally, I do not think it should be possible for a measure with any (and therefore infinitely many) infinite integer moments to be determined by its integer moments. Intuitively, this is because we can essentially choose the power of x which is the first unbounded moment and then construct two measures with the same integer moments, but different values on the moments between the last bounded integer moment and the first unbounded one. I do not see how to make this rigorous though.

A careful phrasing of the question I mean to ask is this: Given a finite set of real numbers $a_n$ which are the finite moments of a probability measure $\mu$ is it always possible to construct a different probability measure $\nu$ for which the only finite moments are $a_n$? It would be nice if it were also possible to say that if a moment of $\mu$ does not exist in the Lebesgue sense, then the corresponding moment of $\nu$ should not exist in the Lebesgue sense either.

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I'm a little confused by the second paragraph. So take a distribution that up to normalization looks like $\frac{1}{1+x^4}$ and a distribution $\frac{1}{a+bx^{9/2}}$ where I have chosen $a,b$ to match the first and second moments of the first distribution. The first distribution has only finite first and second moments, whereas the second distribution has a third moment. Is this the construction you are looking for? –  Alex R. Apr 6 '12 at 19:11
    
@Sam The second paragraph is purely heuristic and I'm not even sure it's very well stated. Ideally I would like to work with probability measures since that's where the question originates. I will edit the question to make it better posed. –  Chris Janjigian Apr 6 '12 at 19:42

1 Answer 1

Going from the comments, why not a probability measure with density proportional to:

$p(x)~\frac{1}{1+x^4}$

and another probability measure proportional to:

$q(x)~\frac{1}{a+bx^{3.9}}$

where $a$ and $b$ are chosen such that the first two moments of $p$ are equal to the first two moments of $q$. Both $p$ and $q$ have only the first two moments finite. As far as "probability measures" go $\mu(A):=\int_A p(x)dx$ and $\nu(A)=\int_Aq(x)dx$ define probability measures on the real line (unless I'm misunderstanding you).

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