When do moments determine a function?
The use of the word moment" in mathematics is related to its use in physics, as in moment arm or moment of inertia. For a non-negative integer n, the nth moment of a function f is the integral of xn f(x) over the function's domain.
UniquenessIf two continuous functions f and g have all the same moments, are they the same function? The answer is yes for functions over a finite interval, but no for functions over an unbounded interval.
ExistenceNow let's consider starting with a set of moments rather than starting with a function. Given a set of moments m0, m1, m2, ... is there a function that has these moments? Typically no.
A better question is what conditions on the moments are necessary for there to exist a function with these moments. This question breaks into three questions
- The Hausdorff moment problem
- The Stieltjes moment problem
- The Hamburger moment problem
depending on whether the function domain is a finite interval, a half-bounded interval, or the real line. For each problem there are known conditions that are necessary and sufficient, but the conditions are different for each problem.
Interestingly, each of the three names Hausdorff, Stieltjes, and Hamburger are well known. Felix Hausdorff is best known for his work in topology: Hausdorff spaces, etc. Thomas Stieltjes is best known for the Riemann-Stieltjes integral, and for his work on continued fractions. Hans Ludwig Hamburger is not as well known, though his last name is certainly familiar.
Finite momentsA practical question in probability is how well a finite number of moments determine a probability distribution. They cannot uniquely determine the distribution, but the do establish bounds for how different the two distributions can be. See this post.
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