Talk:Moment problem

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Maybe redirect the positive definite link to Positive-definite matrix ? elpincha 16:08, 16 Feb 2005 (UTC)

Go for it. Charles Matthews 19:01, 16 Feb 2005 (UTC)

I did, and I met an error while doing it. Looks like a software bug. The Preview shows it correctly and then the main article page does not. Is this a newbie trap or anything? elpincha 20:12, 16 Feb 2005 (UTC)

Most likely it is just the database sending a cached version. Strange things often happen when the servers are somewhat overloaded. Best to keep a copy of any long edits, though, since a proportion do just get lost. Charles Matthews 21:17, 16 Feb 2005 (UTC)

OK Charles. After a few hours, the Preview page still showed the change but the main one did not. First time this happens to me. I was beginning to think that there was a bug whereby, when you use a pipe character within a link (double-square-brackets) and the term to right of the pipe can be linked in of itelf, the software will look into that part. But subsequent experiments proved me wrong, and now I believe some caching stuff happens, just like you say. Anyway, Thanks. elpincha 08:31, 17 Feb 2005 (UTC)

defined domain[edit]

at least, for known distribution, we could find out the set {t: g(t)<infinite} if the set is not R. where g(t) is the moment generating function. Jackzhp (talk) 15:30, 21 January 2009 (UTC)[reply]

How-to[edit]

This article explains when there is a solution, when it is unique and all that, but it does not give a clue as how to actually find such a solution, and neither do the other articles about more specific moment problems. Would anybody know of what are standard techniques, or if such exist at all? Greets, MuDavid (talk) 11:03, 19 October 2010 (UTC)[reply]

Very late answer but since it's a valid question I figured I will try to address it.

It's a very good question that is very nontrivial.

As long as a full sequence of moments is specified, the resulting measure would be unique by the Stone-Weierstrass density argument as stated in the article. Now if you're only given finitely many moments, say, {x₀, x₁, x₂} such that the Hankel matrix

B={\begin{pmatrix}x_{0}&x_{1}\\x_{1}&x_{2}\\\end{pmatrix}}

is positive definite, the solution is not unique. I believe the article calls this the "truncated moment problem". In operator-theoretic terms(see Hamburger moment problem), identifying all possible solutions means identifying all possible extensions of certain symmetric operator. This in turn translates to the following problem: pretend the positive matrix B sits at the upper left corner of a infinite-by-infinite matrix B', how many ways can you fill out the remaining entries of this B' so that the resulting B' is still Hankel and positive definite? Not an easy problem at all. In operator theory, this leads to what is called the Jacobi parametrization of positive matrices.

An analogous situation happens for moment problems on the circle (see Bochner's theorem). There the answer is tied to the positivity of certain Toeplitz matrix (Toeplitz is kind of like mirror image of Hankel), and consequently unitary extensions of certain partial isometry. And it comes down to exactly the same question: given a 2 by 2 positive Toeplitz matrix, how many ways can you extend it to an infinite by infinite positive Toeplitz matrix? This leads to what it called the Schur parametrization of positive matrices.

It turns out somehow the Toeplitz case is easier than the Hankel case to parametrize. To consider the unitary extensions of a partial isometry, you just look at its Fredholm index. Extensions of symmetric operators are more involved.

So the short answer is there is no easy answer; a lot of nontrivial research has been done looking into this issue. Others may have a different perspective. Mct mht (talk) 00:05, 6 September 2012 (UTC)[reply]

Your answer has, alas, nothing to do with my question, even though it does deal with another problem that might potentially be interesting in its own right. What I'm asking is: given a series of moments corresponding to some (unknown) measure μ, how do I find μ? All these musing about the existence and uniqueness of solutions and about how many ways there are to extend an incomplete problem to a valid complete one are all fine if one is an ivory-tower mathematician with no interest in the real world whatsoever. But for us, mortal humans, the only thing that matters is the value of μ. So, given all the moments, what is the value of μ? MuDavid (talk) 17:47, 30 July 2013 (UTC)[reply]